{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,19]],"date-time":"2026-03-19T19:12:38Z","timestamp":1773947558541,"version":"3.50.1"},"reference-count":41,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2022,8,16]],"date-time":"2022-08-16T00:00:00Z","timestamp":1660608000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11575088"],"award-info":[{"award-number":["11575088"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The recently proposed restricted phase space thermodynamics is shown to be applicable to a large class of higher dimensional higher curvature gravity models coupled to Maxwell field, which are known as black hole scan models and are labeled by the spacetime dimension d and the highest order k of the Lanczos-Lovelock densities appearing in the action. Three typical example cases with (d,k)=(5,1),(5,2) and (6,2) are chosen as example cases and studied in some detail. These cases are representatives of Einstein-Hilbert, Chern-Simons and Born-Infield like gravity models. Our study indicates that the Einstein-Hilbert and Born-Infield like gravity models have similar thermodynamic behaviors, e.g., the existence of isocharge T\u2212S phase transitions with the same critical exponents, the existence of isovoltage T\u2212S transitions and the Hawking-Page like transitions, and the similar high temperature asymptotic behaviors for the isocharge heat capacities, etc. However, the Chern-Simons like (5,2)-model behaves quite differently. Neither isocharge nor isovoltage T\u2212S transitions could occur and no Hawking-Page like transition is allowed. This seems to indicate that the Einstein-Hilbert and Born-Infield like models belong to the same universality class while the Chern-Simons like models do not.<\/jats:p>","DOI":"10.3390\/e24081131","type":"journal-article","created":{"date-parts":[[2022,8,16]],"date-time":"2022-08-16T23:44:25Z","timestamp":1660693465000},"page":"1131","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Restricted Phased Space Thermodynamics for Black Holes in Higher Dimensions and Higher Curvature Gravities"],"prefix":"10.3390","volume":"24","author":[{"given":"Xiangqing","family":"Kong","sequence":"first","affiliation":[{"name":"School of Physics, Nankai University, Tianjin 300071, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5830-9826","authenticated-orcid":false,"given":"Tao","family":"Wang","sequence":"additional","affiliation":[{"name":"School of Physics, Nankai University, Tianjin 300071, China"}]},{"given":"Zeyuan","family":"Gao","sequence":"additional","affiliation":[{"name":"School of Physics, Nankai University, Tianjin 300071, China"}]},{"given":"Liu","family":"Zhao","sequence":"additional","affiliation":[{"name":"School of Physics, Nankai University, Tianjin 300071, China"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,16]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"737","DOI":"10.1007\/BF02757029","article-title":"Black holes and the second law","volume":"4","author":"Bekenstein","year":"1972","journal-title":"Lett. Nuovo Cim."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2333","DOI":"10.1103\/PhysRevD.7.2333","article-title":"Black holes and entropy","volume":"7","author":"Bekenstein","year":"1973","journal-title":"Phys. Rev. D"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1007\/BF01645742","article-title":"The four laws of black hole mechanics","volume":"31","author":"Bardeen","year":"1973","journal-title":"Comm. Math. Phys."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1007\/BF02345020","article-title":"Particle creation by black holes","volume":"43","author":"Hawking","year":"1975","journal-title":"Comm. Math. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"195011","DOI":"10.1088\/0264-9381\/26\/19\/195011","article-title":"Enthalpy and the mechanics of AdS black holes","volume":"26","author":"Kastor","year":"2009","journal-title":"Class. Quant. Grav."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"125020","DOI":"10.1088\/0264-9381\/28\/12\/125020","article-title":"The cosmological constant and the black hole equation of state","volume":"28","author":"Dolan","year":"2010","journal-title":"Class. Quant. Grav."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"235017","DOI":"10.1088\/0264-9381\/28\/23\/235017","article-title":"Pressure and volume in the first law of black hole thermodynamics","volume":"28","author":"Dolan","year":"2011","journal-title":"Class. Quant. Grav."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"127503","DOI":"10.1103\/PhysRevD.84.127503","article-title":"Compressibility of rotating black holes","volume":"84","author":"Dolan","year":"2011","journal-title":"Phys. Rev. D"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"33","DOI":"10.1007\/JHEP07(2012)033","article-title":"P-V criticality of charged AdS black holes","volume":"2012","author":"Mann","year":"2012","journal-title":"J. High Energy Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/JHEP09(2013)005","article-title":"PV criticality in the extended phase space of Gauss-Bonnet black holes in AdS space","volume":"2013","author":"Cai","year":"2013","journal-title":"J. High Energy Phys."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"063001","DOI":"10.1088\/1361-6382\/aa5c69","article-title":"Black hole chemistry: Thermodynamics with Lambda","volume":"34","author":"Mann","year":"2017","journal-title":"Class. Quant. Grav."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"214","DOI":"10.1016\/j.physletb.2014.07.019","article-title":"Critical phenomena of static charged AdS black holes in conformal gravity","volume":"736","author":"Xu","year":"2014","journal-title":"Phys. Lett. B"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"2970","DOI":"10.1140\/epjc\/s10052-014-2970-8","article-title":"Gauss\u2013Bonnet coupling constant as a free thermodynamical variable and the associated criticality","volume":"74","author":"Xu","year":"2014","journal-title":"Eur. Phys. J. C"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"296","DOI":"10.1016\/j.physletb.2018.08.075","article-title":"Black hole thermodynamics with the cosmological constant as independent variable: Bridge between the enthalpy and the Euclidean path integral approaches","volume":"786","author":"Lemos","year":"2018","journal-title":"Phys. Lett. B"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"044028","DOI":"10.1103\/PhysRevD.91.044028","article-title":"Phase transition and thermodynamical geometry of Reissner-Nordstr\u00f6m-AdS black holes in extended phase space","volume":"91","author":"Zhang","year":"2015","journal-title":"Phys. Rev. D"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"106014","DOI":"10.1103\/PhysRevD.105.106014","article-title":"Holographic thermodynamics requires a chemical potential for color","volume":"105","author":"Visser","year":"2022","journal-title":"Phys. Rev. D"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/JHEP12(2015)073","article-title":"Holographic black hole chemistry","volume":"2015","author":"Karch","year":"2015","journal-title":"J. High Energy Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"091301","DOI":"10.1103\/PhysRevLett.127.091301","article-title":"Thermodynamics of AdS black holes: Critical behavior of the central charge","volume":"127","author":"Cong","year":"2021","journal-title":"Phys. Rev. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1103\/PhysRevLett.30.71","article-title":"Mass formula for Kerr black holes","volume":"30","author":"Smarr","year":"1973","journal-title":"Phys. Rev. Lett."},{"key":"ref_20","unstructured":"Callen, H.B. (1985). Thermodynamics and an Introduction to Thermostatistics, John Willy & Sons. [2nd ed.]."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"075019","DOI":"10.1088\/1361-6382\/ac566c","article-title":"Restricted phase space thermodynamics for AdS black holes via holography","volume":"39","author":"Gao","year":"2021","journal-title":"Class. Quant. Grav."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1140\/epjc\/s10052-022-10080-y","article-title":"Thermodynamics of Kerr-AdS black holes in the restricted phase space","volume":"82","author":"Gao","year":"2022","journal-title":"Eur. Phys. J. C"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"136935","DOI":"10.1016\/j.physletb.2022.136935","article-title":"Black hole thermodynamics is extensive with variable newton constant","volume":"827","author":"Wang","year":"2022","journal-title":"Phys. Lett. B"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"055105","DOI":"10.1088\/1674-1137\/ac4f4c","article-title":"Thermodynamics for higher dimensional rotating black holes with variable Newton constant","volume":"46","author":"Zhao","year":"2022","journal-title":"Chin. Phys. C"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2752","DOI":"10.1103\/PhysRevD.15.2752","article-title":"Action integrals and partition functions in quantum gravity","volume":"15","author":"Gibbons","year":"1977","journal-title":"Phys. Rev. D"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"2092","DOI":"10.1103\/PhysRevD.33.2092","article-title":"Black-hole thermodynamics and the Euclidean Einstein action","volume":"33","author":"York","year":"1986","journal-title":"Phys. Rev. D"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1503","DOI":"10.1088\/0264-9381\/22\/9\/002","article-title":"The first law of thermodynamics for Kerr\u2013anti-de Sitter black holes","volume":"22","author":"Gibbons","year":"2005","journal-title":"Class. Quant. Grav."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1007\/JHEP10(2014)170","article-title":"Holographic Entropy Production","volume":"1410","author":"Tian","year":"2014","journal-title":"J. High Energy Phys."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"245001","DOI":"10.1088\/1361-6382\/ab5343","article-title":"A topological charge of black holes","volume":"36","author":"Tian","year":"2019","journal-title":"Class. Quantum Grav."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"084014","DOI":"10.1103\/PhysRevD.65.084014","article-title":"Gauss-bonnet black holes in AdS spaces","volume":"65","author":"Cai","year":"2002","journal-title":"Phys. Rev. D"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"147","DOI":"10.1007\/BF01351210","article-title":"Elektromagnetismus als nat\u00fcrliche eigenschaft der riemannschen geometrie","volume":"73","author":"Lanczos","year":"1932","journal-title":"Z. Phys."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"498","DOI":"10.1063\/1.1665613","article-title":"The Einstein tensor and its generalizations","volume":"12","author":"Lovelock","year":"1971","journal-title":"J. Math. Phys."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"084013","DOI":"10.1103\/PhysRevD.62.084013","article-title":"Black hole scan","volume":"62","author":"Crisostomo","year":"2000","journal-title":"Phys. Rev. D"},{"key":"ref_34","unstructured":"Wu, B. (2022, January 3\u20138). Holographic thermodynamics of BTZ black hole. Proceedings of the 2022 Annual Meeting of Chinese Association of Gravitation and Astrophysics, Beijing, China. Paper Version to Appear."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"R3427","DOI":"10.1103\/PhysRevD.48.R3427","article-title":"Black hole entropy is the Noether charge","volume":"48","author":"Wald","year":"1993","journal-title":"Phys. Rev. D"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1103\/PhysRevD.50.846","article-title":"Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy","volume":"50","author":"Iyer","year":"1994","journal-title":"Phys. Rev. D"},{"key":"ref_37","first-page":"4","article-title":"The stability of classical solutions","volume":"24","year":"1976","journal-title":"Sov. J. Nucl. Phys."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/BF01208266","article-title":"Thermodynamics of black holes in anti-de Sitter space","volume":"87","author":"Hawking","year":"1983","journal-title":"Commun. Math. Phys."},{"key":"ref_39","unstructured":"Tong, D. (2022, March 01). Lectures on Statistical Physics. Notes Online from the Author\u2019s DAMTP Teaching Page. Available online: https:\/\/www.damtp.cam.ac.uk\/user\/tong\/statphys.html."},{"key":"ref_40","unstructured":"Thompson, C.J. (1988). Classical Equilibrium Statistical Mechanics, Oxford University Press."},{"key":"ref_41","unstructured":"Pathria, R.K., and Beale, P.D. (2011). Statistical Mechanics, Elsevier. [3rd ed.]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1131\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T00:09:16Z","timestamp":1760141356000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/8\/1131"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,8,16]]},"references-count":41,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2022,8]]}},"alternative-id":["e24081131"],"URL":"https:\/\/doi.org\/10.3390\/e24081131","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,8,16]]}}}