Horizon entropy consistent with FLRW equations for general modified theories of gravity and for all EoS of the matter field

Understand

The question that continues to hinge the interrelation between cosmology and thermodynamics is broadly described as -- what is the form of horizon entropy that links the Friedmann equations for a "$general$" gravity theory with the underlying thermodynamics of the apparent horizon? The answer to this question was known only for Einstein's gravity and for $(n+1)$ dimensional Gauss-Bonnet gravity theory, but not for a general modified theory of gravity (for instance, the $F(R)$ gravity).

• In the present work, we take this issue and determine a general form of entropy that connects the Friedmann equations for any gravity theory with the apparent horizon thermodynamics given by $TdS = -dE + WdV$ (the symbols have their usual meaning in the context of entropic cosmology and $W = \left(\rho - p\right)/2$ is the work density of the matter fields represented by $\rho$ and $p$ as the energy density and the pressure, respectively).
• Using such generalized entropy, we find the respective entropies for several modified theories of gravity (including the $F(R)$ gravity).
• Further, it turns out that besides the above-mentioned question, the thermodynamic law $TdS = -dE + WdV$ itself has some serious difficulties for certain values of $\omega$ (the EoS of matter field).