{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T16:49:11Z","timestamp":1774543751933,"version":"3.50.1"},"reference-count":44,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2019,2,13]],"date-time":"2019-02-13T00:00:00Z","timestamp":1550016000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We study thermodynamics in     f ( R )     gravity with the disformal transformation. The transformation applied to the matter Lagrangian has the form of      \u03b3  \u03bc \u03bd   = A  ( \u03d5 , X )   g  \u03bc \u03bd   + B  ( \u03d5 , X )   \u2202 \u03bc  \u03d5  \u2202 \u03bd  \u03d5     with the assumption of the Minkowski matter metric      \u03b3  \u03bc \u03bd   =  \u03b7  \u03bc \u03bd      , where    \u03d5    is the disformal scalar and X is the corresponding kinetic term of    \u03d5   . We verify the generalized first and second laws of thermodynamics in this disformal type of     f ( R )     gravity in the Friedmann-Lema\u00eetre-Robertson-Walker (FLRW) universe. In addition, we show that the Hubble parameter contains the disformally induced terms, which define the effectively varying equations of state for matter.<\/jats:p>","DOI":"10.3390\/e21020172","type":"journal-article","created":{"date-parts":[[2019,2,14]],"date-time":"2019-02-14T03:21:46Z","timestamp":1550114506000},"page":"172","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Thermodynamics of f(R) Gravity with Disformal Transformation"],"prefix":"10.3390","volume":"21","author":[{"given":"Chao-Qiang","family":"Geng","sequence":"first","affiliation":[{"name":"Synergetic Innovation Center for Quantum Effects and Applications (SICQEA), Hunan Normal University, Changsha 410081, China"},{"name":"Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan"},{"name":"Physics Division, National Center for Theoretical Sciences, Hsinchu 30013, Taiwan"}]},{"given":"Wei-Cheng","family":"Hsu","sequence":"additional","affiliation":[{"name":"Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan"}]},{"given":"Jhih-Rong","family":"Lu","sequence":"additional","affiliation":[{"name":"Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan"}]},{"given":"Ling-Wei","family":"Luo","sequence":"additional","affiliation":[{"name":"Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan"},{"name":"Institute of Physics, Academia Sinica, Taipei 11529, Taiwan"}]}],"member":"1968","published-online":{"date-parts":[[2019,2,13]]},"reference":[{"key":"ref_1","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. 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