Modified gravity as entropic cosmology

The present work reveals a direct correspondence between modified theories of gravity (cosmology) and entropic cosmology based on the thermodynamics of apparent horizon. It turns out that due to the total differentiable property of entropy, the usual thermodynamic law (used for Einstein gravity) needs to be generalized for modified gravity theories having more than one thermodynamic degrees of freedom (d.o.f.). For the modified theories having $n$ number of thermodynamic d.o.f., the corresponding horizon entropy is given by: $S_\mathrm{h} \sim S_\mathrm{BH} +$ terms containing the time derivatives of $S_\mathrm{BH}$ up to $(n-1)$-th order, and moreover, the coefficient(s) of the derivative term(s) are proportional to the modification parameter of the gravity theory (compared to the Einstein gravity; $S_\mathrm{BH}$ is the Bekenstein-Hawking entropy). By identifying the independent thermodynamic variables from the first law of thermodynamics, we show that the equivalent thermodynamic description of modified gravity naturally allows the time derivative of the Bekenstein-Hawking entropy in the horizon entropy.