{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,18]],"date-time":"2026-01-18T04:39:56Z","timestamp":1768711196838,"version":"3.49.0"},"reference-count":116,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,20]],"date-time":"2025-08-20T00:00:00Z","timestamp":1755648000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU)","award":["IMSIU DDRSP2502"],"award-info":[{"award-number":["IMSIU DDRSP2502"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this work, we explore a class of spherically symmetric black hole (BH) solutions within the framework of modified gravity, focusing on a non-ghost-free f(R,G) theory coupled to a scalar field. We present a novel black hole geometry that arises as a deformation of the Schwarzschild solution and analyze its physical and thermodynamic properties. Our results show that the model satisfies stability conditions, with the Ricci scalar R, as well as its first and second derivatives, remaining positive throughout the spacetime. The solution admits multiple horizons and exhibits strong curvature singularities compared to those in general relativity. Furthermore, it supports a non-trivial scalar field potential. A comprehensive thermodynamic analysis is performed, including evaluations of the entropy, temperature, heat capacity, and quasi-local energy. We find that the black hole exhibits thermodynamic stability within certain ranges of model parameters. In addition, we investigate geodesic deviation and derive the conditions necessary for stability within the f(R,G) gravitational framework.<\/jats:p>","DOI":"10.3390\/sym17081360","type":"journal-article","created":{"date-parts":[[2025,8,20]],"date-time":"2025-08-20T07:50:46Z","timestamp":1755676246000},"page":"1360","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Black Hole Solution in f(R,G) Gravitational Theory Coupled with Scalar Field"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5544-1119","authenticated-orcid":false,"given":"G. G. L.","family":"Nashed","sequence":"first","affiliation":[{"name":"Centre for Theoretical Physics, The British University in Egypt, P.O. Box 43, El Sherouk City 11837, Cairo, Egypt"},{"name":"Centre for Space Research, North-West University, Potchefstroom 2531, South Africa"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4189-0906","authenticated-orcid":false,"given":"A.","family":"Eid","sequence":"additional","affiliation":[{"name":"Department of Physics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11564, Saudi Arabia"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,20]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"191101","DOI":"10.1103\/PhysRevLett.120.191101","article-title":"New General Relativistic Contribution to Mercury\u2019s Perihelion Advance","volume":"120","author":"Will","year":"2018","journal-title":"Phys. Rev. Lett."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"124001","DOI":"10.1088\/0264-9381\/32\/12\/124001","article-title":"The 1919 measurement of the deflection of light","volume":"32","author":"Will","year":"2015","journal-title":"Class. Quant. Grav."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"061102","DOI":"10.1103\/PhysRevLett.116.061102","article-title":"Observation of Gravitational Waves from a Binary Black Hole Merger","volume":"116","author":"Abbott","year":"2016","journal-title":"Phys. Rev. Lett."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"161101","DOI":"10.1103\/PhysRevLett.119.161101","article-title":"GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral","volume":"119","author":"Abbott","year":"2017","journal-title":"Phys. Rev. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"L1","DOI":"10.3847\/2041-8213\/ab0ec7","article-title":"First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole","volume":"875","author":"Akiyama","year":"2019","journal-title":"Astrophys. J. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"L2","DOI":"10.3847\/2041-8213\/ab0c96","article-title":"First M87 Event Horizon Telescope Results. II. Array and Instrumentation","volume":"875","author":"Akiyama","year":"2019","journal-title":"Astrophys. J. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"L3","DOI":"10.3847\/2041-8213\/ab0c57","article-title":"First M87 Event Horizon Telescope Results. III. Data Processing and Calibration","volume":"875","author":"Akiyama","year":"2019","journal-title":"Astrophys. J. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"L4","DOI":"10.3847\/2041-8213\/ab0e85","article-title":"First M87 Event Horizon Telescope Results. IV. Imaging the Central Supermassive Black Hole","volume":"875","author":"Akiyama","year":"2019","journal-title":"Astrophys. J. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"L5","DOI":"10.3847\/2041-8213\/ab0f43","article-title":"First M87 Event Horizon Telescope Results. V. Physical Origin of the Asymmetric Ring","volume":"875","author":"Akiyama","year":"2019","journal-title":"Astrophys. J. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"L6","DOI":"10.3847\/2041-8213\/ab1141","article-title":"First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole","volume":"875","author":"Akiyama","year":"2019","journal-title":"Astrophys. J. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"L12","DOI":"10.3847\/2041-8213\/abe71d","article-title":"First M87 Event Horizon Telescope Results. VII. Polarization of the Ring","volume":"910","author":"Akiyama","year":"2021","journal-title":"Astrophys. J. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"L13","DOI":"10.3847\/2041-8213\/abe4de","article-title":"First M87 Event Horizon Telescope Results. VIII. Magnetic Field Structure near The Event Horizon","volume":"910","author":"Akiyama","year":"2021","journal-title":"Astrophys. J. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"L20","DOI":"10.3847\/2041-8213\/acff70","article-title":"First M87 Event Horizon Telescope Results. IX. Detection of Near-horizon Circular Polarization","volume":"957","author":"Akiyama","year":"2023","journal-title":"Astrophys. J. Lett."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"L12","DOI":"10.3847\/2041-8213\/ac6674","article-title":"First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way","volume":"930","author":"Akiyama","year":"2022","journal-title":"Astrophys. J. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"L13","DOI":"10.3847\/2041-8213\/ac6675","article-title":"First Sagittarius A* Event Horizon Telescope Results. II. EHT and Multiwavelength Observations, Data Processing, and Calibration","volume":"930","author":"Akiyama","year":"2022","journal-title":"Astrophys. J. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"L14","DOI":"10.3847\/2041-8213\/ac6429","article-title":"First Sagittarius A* Event Horizon Telescope Results. III. Imaging of the Galactic Center Supermassive Black Hole","volume":"930","author":"Akiyama","year":"2022","journal-title":"Astrophys. J. Lett."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"L15","DOI":"10.3847\/2041-8213\/ac6736","article-title":"First Sagittarius A* Event Horizon Telescope Results. IV. Variability, Morphology, and Black Hole Mass","volume":"930","author":"Akiyama","year":"2022","journal-title":"Astrophys. J. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"L16","DOI":"10.3847\/2041-8213\/ac6672","article-title":"First Sagittarius A* Event Horizon Telescope Results. V. Testing Astrophysical Models of the Galactic Center Black Hole","volume":"930","author":"Akiyama","year":"2022","journal-title":"Astrophys. J. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"L17","DOI":"10.3847\/2041-8213\/ac6756","article-title":"First Sagittarius A* Event Horizon Telescope Results. VI. Testing the Black Hole Metric","volume":"930","author":"Akiyama","year":"2022","journal-title":"Astrophys. J. Lett."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1086\/307221","article-title":"Measurements of Omega and Lambda from 42 High-Redshift Supernovae","volume":"517","author":"Perlmutter","year":"1999","journal-title":"Astrophys. J."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"1009","DOI":"10.1086\/300499","article-title":"Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant","volume":"116","author":"Riess","year":"1998","journal-title":"Astron. J."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1086\/306495","article-title":"Supernova limits on the cosmic equation of state","volume":"509","author":"Garnavich","year":"1998","journal-title":"Astrophys. J."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"451","DOI":"10.1103\/RevModPhys.82.451","article-title":"f(R) Theories of Gravity","volume":"82","author":"Sotiriou","year":"2010","journal-title":"Rev. Mod. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2012.01.001","article-title":"Modified Gravity and Cosmology","volume":"513","author":"Clifton","year":"2012","journal-title":"Phys. Rept."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.1142\/S021827180600942X","article-title":"Dynamics of Dark Energy","volume":"15","author":"Copeland","year":"2006","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"525","DOI":"10.1088\/0253-6102\/56\/3\/24","article-title":"Dark Energy","volume":"56","author":"Li","year":"2011","journal-title":"Commun. Theor. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"123512","DOI":"10.1103\/PhysRevD.68.123512","article-title":"Modified Gravity with Negative and Positive Powers of the Curvature: Unification of the Inflation and of the Cosmic Acceleration","volume":"68","author":"Nojiri","year":"2003","journal-title":"Phys. Rev. D"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1016\/j.physletb.2003.09.091","article-title":"Where new gravitational physics comes from: M Theory?","volume":"576","author":"Nojiri","year":"2003","journal-title":"Phys. Lett. B"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"086005","DOI":"10.1103\/PhysRevD.74.086005","article-title":"Modified f(R) gravity consistent with realistic cosmology: From a matter dominated epoch to a dark energy universe","volume":"74","author":"Nojiri","year":"2006","journal-title":"Phys. Rev. D"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"2050090","DOI":"10.1142\/S021827182050090X","article-title":"Thermodynamical correspondence of f(R) gravity in the Jordan and Einstein frames","volume":"29","author":"Nashed","year":"2020","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1142\/S0219887807001928","article-title":"Introduction to modified gravity and gravitational alternative for dark energy","volume":"4","author":"Nojiri","year":"2007","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"74","DOI":"10.1016\/j.physletb.2009.09.045","article-title":"Cosmological reconstruction of realistic modified F(R) gravities","volume":"681","author":"Nojiri","year":"2009","journal-title":"Phys. Lett. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/j.physrep.2011.04.001","article-title":"Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models","volume":"505","author":"Nojiri","year":"2011","journal-title":"Phys. Rept."},{"key":"ref_34","first-page":"521","article-title":"Vacuum nonsingular black hole in tetrad theory of gravitation","volume":"117","author":"Nashed","year":"2002","journal-title":"Nuovo Cim. B"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1016\/j.physletb.2006.06.034","article-title":"Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phase","volume":"639","author":"Capozziello","year":"2006","journal-title":"Phys. Lett. B"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Cognola, G., Elizalde, E., Nojiri, S., Odintsov, S.D., Sebastiani, L., and Zerbini, S. (2008). A Class of viable modified f(R) gravities describing inflation and the onset of accelerated expansion. Phys. Rev. D, 77.","DOI":"10.1103\/PhysRevD.77.046009"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"3","DOI":"10.12942\/lrr-2010-3","article-title":"f(R) theories","volume":"13","author":"Tsujikawa","year":"2010","journal-title":"Living Rev. Rel."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/j.physrep.2011.09.003","article-title":"Extended Theories of Gravity","volume":"509","author":"Capozziello","year":"2011","journal-title":"Phys. Rept."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"007","DOI":"10.1088\/1475-7516\/2021\/11\/007","article-title":"Rotating black hole in f(R) theory","volume":"11","author":"Nashed","year":"2021","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"124054","DOI":"10.1103\/PhysRevD.104.124054","article-title":"Specific neutral and charged black holes in f(R) gravitational theory","volume":"104","author":"Nashed","year":"2021","journal-title":"Phys. Rev. D"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"136475","DOI":"10.1016\/j.physletb.2021.136475","article-title":"Analytic charged BHs in f(R) gravity","volume":"820","author":"Nashed","year":"2021","journal-title":"Phys. Lett. B"},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"704","DOI":"10.1140\/epjc\/s10052-020-8258-2","article-title":"f(G) Noether cosmology","volume":"80","author":"Bajardi","year":"2020","journal-title":"Eur. Phys. J. C"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"083531","DOI":"10.1103\/PhysRevD.91.083531","article-title":"Cosmological inflation in F(R,G) gravity","volume":"91","author":"Paolella","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_44","doi-asserted-by":"crossref","unstructured":"Bajardi, F., Dialektopoulos, K.F., and Capozziello, S. (2020). Higher Dimensional Static and Spherically Symmetric Solutions in Extended Gauss\u2013Bonnet Gravity. Symmetry, 12.","DOI":"10.3390\/sym12030372"},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1007\/s10509-016-2786-0","article-title":"Reconstruction of f(T)-gravity in the absence of matter","volume":"361","author":"Nashed","year":"2016","journal-title":"Astrophys. Space Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"1450164","DOI":"10.1142\/S0217732314501648","article-title":"Noether Symmetry Approach in Gauss-Bonnet Cosmology","volume":"29","author":"Capozziello","year":"2014","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1140\/epjp\/i2015-15124-3","article-title":"Kerr-NUT black hole thermodynamics in f(T) gravity theories","volume":"130","author":"Nashed","year":"2015","journal-title":"Eur. Phys. J. Plus"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"124027","DOI":"10.1103\/PhysRevD.92.124027","article-title":"Singular Bouncing Cosmology from Gauss-Bonnet Modified Gravity","volume":"92","author":"Oikonomou","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.physrep.2017.06.001","article-title":"Modified gravity theories on a nutshell: Inflation, bounce and late-time evolution","volume":"692","author":"Nojiri","year":"2017","journal-title":"Phys. Rept."},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"153","DOI":"10.3847\/0004-637X\/827\/2\/153","article-title":"NONSINGULAR UNIVERSES IN GAUSS\u2013BONNET GRAVITY\u2019S RAINBOW","volume":"827","author":"Hendi","year":"2016","journal-title":"Astrophys. J."},{"key":"ref_51","doi-asserted-by":"crossref","first-page":"064039","DOI":"10.1103\/PhysRevD.100.064039","article-title":"Spontaneous scalarization of Gauss-Bonnet black holes: Analytic treatment in the linearized regime","volume":"100","author":"Hod","year":"2019","journal-title":"Phys. Rev. D"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"135535","DOI":"10.1016\/j.physletb.2020.135535","article-title":"Bounds on photon spheres and shadows of charged black holes in Einstein-Gauss-Bonnet-Maxwell gravity","volume":"807","author":"Ma","year":"2020","journal-title":"Phys. Lett. B"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"046005","DOI":"10.1103\/PhysRevD.101.046005","article-title":"Thermodynamic geometry for charged Gauss-Bonnet black holes in AdS spacetimes","volume":"101","author":"Ghosh","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"135437","DOI":"10.1016\/j.physletb.2020.135437","article-title":"Swampland implications of GW170817-compatible Einstein-Gauss-Bonnet gravity","volume":"805","author":"Odintsov","year":"2020","journal-title":"Phys. Lett. B"},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"115135","DOI":"10.1016\/j.nuclphysb.2020.115135","article-title":"Rectifying Einstein-Gauss-Bonnet Inflation in View of GW170817","volume":"958","author":"Odintsov","year":"2020","journal-title":"Nucl. Phys. B"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"024086","DOI":"10.1103\/PhysRevD.102.024086","article-title":"Polar quasinormal modes of the scalarized Einstein-Gauss-Bonnet black holes","volume":"102","author":"Doneva","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"612","DOI":"10.1140\/epjc\/s10052-020-8176-3","article-title":"Generalization of cosmological attractor approach to Einstein\u2013Gauss\u2013Bonnet gravity","volume":"80","author":"Pozdeeva","year":"2020","journal-title":"Eur. Phys. J. C"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"024070","DOI":"10.1103\/PhysRevD.102.024070","article-title":"Kunz, J.; Kanti, P. Properties of ultracompact particlelike solutions in Einstein-scalar-Gauss-Bonnet theories","volume":"102","author":"Kleihaus","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1007\/JHEP08(2020)100","article-title":"Gravitational phase transition mediated by thermalon in Einstein-Gauss-Bonnet-Maxwell-Kalb-Ramond gravity","volume":"2020","author":"Samart","year":"2020","journal-title":"J. High Energy Phys."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"084051","DOI":"10.1103\/PhysRevD.101.084051","article-title":"Thermodynamics of Gauss-Bonnet-de Sitter Black Holes","volume":"101","author":"Haroon","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"232","DOI":"10.1140\/epjc\/s10052-020-7797-x","article-title":"Effect of magnetic field on holographic insulator\/superconductor phase transition in higher dimensional Gauss\u2013Bonnet gravity","volume":"80","author":"Parai","year":"2020","journal-title":"Eur. Phys. J. C"},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"115269","DOI":"10.1016\/j.nuclphysb.2020.115269","article-title":"Charged 4D Einstein-Gauss-Bonnet-AdS black holes: Shadow, energy emission, deflection angle and heat engine","volume":"961","author":"Jafarzade","year":"2020","journal-title":"Nucl. Phys. B"},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"045013","DOI":"10.1103\/PhysRevD.62.045013","article-title":"Effective Gauss-Bonnet interaction in Randall-Sundrum compactification","volume":"62","author":"Kim","year":"2000","journal-title":"Phys. Rev. D"},{"key":"ref_64","doi-asserted-by":"crossref","first-page":"024044","DOI":"10.1103\/PhysRevD.66.024044","article-title":"Anti-de Sitter black holes, thermal phase transition and holography in higher curvature gravity","volume":"66","author":"Cho","year":"2002","journal-title":"Phys. Rev. D"},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"10001","DOI":"10.1209\/0295-5075\/105\/10001","article-title":"Schwarzschild solution in extended teleparallel gravity","volume":"105","author":"Nashed","year":"2014","journal-title":"Europhys. Lett."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"4671","DOI":"10.1088\/0264-9381\/19\/18\/304","article-title":"General Gauss-Bonnet brane cosmology","volume":"19","author":"Charmousis","year":"2002","journal-title":"Class. Quant. Grav."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"066","DOI":"10.1088\/1126-6708\/2003\/10\/066","article-title":"Brane cosmology with curvature corrections","volume":"10","author":"Kofinas","year":"2003","journal-title":"J. High Energy Phys."},{"key":"ref_68","doi-asserted-by":"crossref","first-page":"017","DOI":"10.1088\/1475-7516\/2022\/05\/017","article-title":"Stability of motion and thermodynamics in charged black holes in f(T) gravity","volume":"2022","author":"Nashed","year":"2022","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_69","doi-asserted-by":"crossref","first-page":"104025","DOI":"10.1103\/PhysRevD.69.104025","article-title":"Gauss-Bonnet black holes in dS spaces","volume":"69","author":"Cai","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"114","DOI":"10.1016\/j.physletb.2004.01.019","article-title":"Gauss-Bonnet black holes at the LHC: Beyond the dimensionality of space","volume":"584","author":"Barrau","year":"2004","journal-title":"Phys. Lett. B"},{"key":"ref_71","doi-asserted-by":"crossref","first-page":"024002","DOI":"10.1103\/PhysRevD.69.024002","article-title":"Covariant gravitational equations on brane world with Gauss-Bonnet term","volume":"69","author":"Maeda","year":"2004","journal-title":"Phys. Rev. D"},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"004","DOI":"10.1088\/1475-7516\/2006\/07\/004","article-title":"Mimicking Lambda with a spin-two ghost condensate","volume":"2006","author":"Tolley","year":"2006","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"064038","DOI":"10.1103\/PhysRevD.76.064038","article-title":"Exact solutions for the Einstein-Gauss-Bonnet theory in five dimensions: Black holes, wormholes and spacetime horns","volume":"76","author":"Dotti","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/s10714-007-0398-2","article-title":"Brane universes with Gauss-Bonnet-induced-gravity","volume":"39","author":"Brown","year":"2007","journal-title":"Gen. Relativ. Gravit."},{"key":"ref_75","doi-asserted-by":"crossref","first-page":"104028","DOI":"10.1103\/PhysRevD.76.104028","article-title":"Brane-World Dynamics in Einstein-Gauss-Bonnet Gravity","volume":"76","author":"Maeda","year":"2007","journal-title":"Phys. Rev. D"},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1007\/978-3-540-88460-6_8","article-title":"Higher order gravity theories and their black hole solutions","volume":"769","author":"Charmousis","year":"2009","journal-title":"Lect. Notes Phys."},{"key":"ref_77","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1016\/j.physletb.2010.01.026","article-title":"Thermodynamics of rotating black branes in Gauss-Bonnet-nonlinear Maxwell gravity","volume":"684","author":"Hendi","year":"2010","journal-title":"Phys. Lett. B"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"063501","DOI":"10.1103\/PhysRevD.89.063501","article-title":"Tensor Perturbations from Brane-World Inflation with Curvature Effects","volume":"89","author":"Liu","year":"2014","journal-title":"Phys. Rev. D"},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"3079","DOI":"10.1140\/epjc\/s10052-014-3079-9","article-title":"Thermodynamic analysis of topological black holes in Gauss-Bonnet gravity with nonlinear source","volume":"74","author":"Hendi","year":"2014","journal-title":"Eur. Phys. J. C"},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"004","DOI":"10.1088\/1475-7516\/2014\/06\/004","article-title":"Mapping the ghost free bigravity into braneworld setup","volume":"2014","author":"Yamashita","year":"2014","journal-title":"J. Cosmol. Astropart. Phys."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"083014","DOI":"10.1103\/PhysRevD.92.083014","article-title":"Rotating black holes in Einstein-Dilaton-Gauss-Bonnet gravity with finite coupling","volume":"92","author":"Maselli","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_82","doi-asserted-by":"crossref","first-page":"086006","DOI":"10.1103\/PhysRevD.88.086006","article-title":"Gauss-Bonnet correction to Holographic thermalization: Two-point functions, circular Wilson loops and entanglement entropy","volume":"88","author":"Li","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/JHEP03(2014)031","article-title":"Holographic thermalization with a chemical potential in Gauss-Bonnet gravity","volume":"2014","author":"Zeng","year":"2014","journal-title":"J. High Energy Phys."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"106010","DOI":"10.1103\/PhysRevD.91.106010","article-title":"Holographic thermalization in Gauss-Bonnet gravity with de Sitter boundary","volume":"91","author":"Zhang","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"L101502","DOI":"10.1103\/PhysRevD.110.L101502","article-title":"Thermodynamics of black holes featuring primary scalar hair","volume":"110","author":"Bakopoulos","year":"2024","journal-title":"Phys. Rev. D"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"064024","DOI":"10.1103\/PhysRevD.109.064024","article-title":"Endowing black holes with beyond-Horndeski primary hair: An exact solution framework for scalarizing in every dimension","volume":"109","author":"Baake","year":"2024","journal-title":"Phys. Rev. D"},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"024044","DOI":"10.1103\/PhysRevD.110.024044","article-title":"Compact objects with primary hair in shift and parity symmetric beyond Horndeski gravities","volume":"110","author":"Bakopoulos","year":"2024","journal-title":"Phys. Rev. D"},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"106","DOI":"10.1007\/JHEP08(2014)106","article-title":"Dressing a Black Hole with a Time-Dependent Galileon","volume":"2014","author":"Babichev","year":"2014","journal-title":"J. High Energy Phys."},{"key":"ref_89","doi-asserted-by":"crossref","first-page":"102212","DOI":"10.1016\/j.newast.2024.102212","article-title":"Deflection angle of light in an black hole with primary scalar hair geometry","volume":"109","author":"Mushtaq","year":"2024","journal-title":"New Astron."},{"key":"ref_90","doi-asserted-by":"crossref","first-page":"2400113","DOI":"10.1002\/prop.202400113","article-title":"F(Q) Gravity with Gauss\u2013Bonnet Corrections: From Early-Time Inflation to Late-Time Acceleration","volume":"72","author":"Nojiri","year":"2024","journal-title":"Fortsch. Phys."},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"024014","DOI":"10.1103\/PhysRevD.108.024014","article-title":"Hayward black hole in scalar-Einstein-Gauss-Bonnet gravity in four dimensions","volume":"108","author":"Nojiri","year":"2023","journal-title":"Phys. Rev. D"},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"044024","DOI":"10.1103\/PhysRevD.106.044024","article-title":"Multihorizons black hole solutions, photon sphere, and perihelion shift in weak ghost-free Gauss-Bonnet theory of gravity","volume":"106","author":"Nashed","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_93","doi-asserted-by":"crossref","first-page":"024039","DOI":"10.1103\/PhysRevD.83.024039","article-title":"Robust approach to f(R) gravity","volume":"83","author":"Jaime","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_94","doi-asserted-by":"crossref","first-page":"124022","DOI":"10.1103\/PhysRevD.102.124022","article-title":"Non-trivial black hole solutions in f(R) gravitational theory","volume":"102","author":"Nashed","year":"2020","journal-title":"Phys. Rev. D"},{"key":"ref_95","doi-asserted-by":"crossref","first-page":"100744","DOI":"10.1016\/j.dark.2020.100744","article-title":"Hairy black holes by gravitational decoupling","volume":"31","author":"Ovalle","year":"2021","journal-title":"Phys. Dark Univ."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"101717","DOI":"10.1016\/j.dark.2024.101717","article-title":"Non-singular black hole by gravitational decoupling and some thermodynamic properties","volume":"46","author":"Misyura","year":"2024","journal-title":"Phys. Dark Univ."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"024039","DOI":"10.1103\/PhysRevD.108.024039","article-title":"Energy extraction via Comisso-Asenjo mechanism from rotating hairy black hole","volume":"108","author":"Li","year":"2023","journal-title":"Phys. Rev. D"},{"key":"ref_98","doi-asserted-by":"crossref","first-page":"168557","DOI":"10.1016\/j.aop.2021.168557","article-title":"Geodesic motion around hairy black holes","volume":"431","author":"Ramos","year":"2021","journal-title":"Ann. Phys."},{"key":"ref_99","unstructured":"Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). Gravitation, W. H. Freeman."},{"key":"ref_100","doi-asserted-by":"crossref","first-page":"101462","DOI":"10.1016\/j.dark.2024.101462","article-title":"Extension of Hayward black hole in f(R) gravity coupled with a scalar field","volume":"44","author":"Nashed","year":"2024","journal-title":"Phys. Dark Univ."},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"023515","DOI":"10.1103\/PhysRevD.84.023515","article-title":"On the Energy Issue for a Class of Modified Higher Order Gravity Black Hole Solutions","volume":"84","author":"Cognola","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"682","DOI":"10.1140\/epjc\/s10052-018-6167-4","article-title":"Horizon thermodynamics in f(R) theory","volume":"78","author":"Zheng","year":"2018","journal-title":"Eur. Phys. J. C"},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"024013","DOI":"10.1103\/PhysRevD.86.024013","article-title":"Higher-dimensional charged f(R) black holes","volume":"86","author":"Sheykhi","year":"2012","journal-title":"Phys. Rev. D"},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1140\/epjc\/s10052-010-1372-9","article-title":"Thermodynamics of apparent horizon and modified Friedmann equations","volume":"69","author":"Sheykhi","year":"2010","journal-title":"Eur. Phys. J. C"},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"703","DOI":"10.1140\/epjc\/s10052-010-1483-3","article-title":"Thermodynamics of higher dimensional topological charged AdS black branes in dilaton gravity","volume":"70","author":"Hendi","year":"2010","journal-title":"Eur. Phys. J. C"},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"084040","DOI":"10.1103\/PhysRevD.81.084040","article-title":"Thermodynamic instability of charged dilaton black holes in AdS spaces","volume":"81","author":"Sheykhi","year":"2010","journal-title":"Phys. Rev. D"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"841","DOI":"10.1016\/S0960-0779(02)00168-6","article-title":"Stability of the vacuum nonsingular black hole","volume":"15","author":"Nashed","year":"2003","journal-title":"Chaos Solitons Fractals"},{"key":"ref_108","doi-asserted-by":"crossref","first-page":"024048","DOI":"10.1103\/PhysRevD.84.024048","article-title":"Instability of rotating black hole in a limited form of f(R) gravity","volume":"84","author":"Myung","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_109","doi-asserted-by":"crossref","first-page":"104017","DOI":"10.1103\/PhysRevD.88.104017","article-title":"Instability of a Kerr black hole in f(R) gravity","volume":"88","author":"Myung","year":"2013","journal-title":"Phys. Rev. D"},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"5917","DOI":"10.1088\/0264-9381\/24\/23\/014","article-title":"Black holes thermodynamics to all order in the Planck length in extra dimensions","volume":"24","author":"Nouicer","year":"2007","journal-title":"Class. Quant. Grav."},{"key":"ref_111","doi-asserted-by":"crossref","unstructured":"Dymnikova, I., and Korpusik, M. (2011). Thermodynamics of Regular Cosmological Black Holes with the de Sitter Interior. Entropy, 13.","DOI":"10.3390\/e13121967"},{"key":"ref_112","doi-asserted-by":"crossref","first-page":"064018","DOI":"10.1103\/PhysRevD.60.064018","article-title":"Charged AdS black holes and catastrophic holography","volume":"60","author":"Chamblin","year":"1999","journal-title":"Phys. Rev. D"},{"key":"ref_113","doi-asserted-by":"crossref","first-page":"687","DOI":"10.1016\/j.physletb.2012.11.017","article-title":"Phase transition of quantum corrected Schwarzschild black hole","volume":"718","author":"Kim","year":"2012","journal-title":"Phys. Lett. B"},{"key":"ref_114","doi-asserted-by":"crossref","unstructured":"D\u2019Inverno, R.A. (1992). Introducing Einstein\u2019s Relativity, Oxford University Press.","DOI":"10.1093\/oso\/9780198596530.001.0001"},{"key":"ref_115","doi-asserted-by":"crossref","first-page":"1136","DOI":"10.1093\/mnras\/stae2302","article-title":"Cosmology in modified f (G) gravity: A late-time cosmic phenomena","volume":"535","author":"Lohakare","year":"2024","journal-title":"Mon. Not. Roy. Astron. Soc."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"215009","DOI":"10.1088\/1361-6382\/acfc0f","article-title":"Observational constrained gravity cosmological model and the dynamical system analysis","volume":"40","author":"Lohakare","year":"2023","journal-title":"Class. Quant. Grav."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1360\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:31:22Z","timestamp":1760034682000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/17\/8\/1360"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,8,20]]},"references-count":116,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2025,8]]}},"alternative-id":["sym17081360"],"URL":"https:\/\/doi.org\/10.3390\/sym17081360","relation":{},"ISSN":["2073-8994"],"issn-type":[{"value":"2073-8994","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,8,20]]}}}