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

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 $F\left(R\right)$ 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 $W=(\rho -p)/2$ 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 $F\left(R\right)$ 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 equations of state (EOSs) of the matter field]. Thus we propose a modified thermodynamic law of apparent horizon, given by $TdS=-dE+\rho dV$, that is interestingly free from such difficulties. The modified law proves to be valid for all EOSs of the matter field and thus is considered to be more general compared to the previous one which, however, is a limiting case of the modified law for $p=-\rho$. Based on such modified thermodynamics, we further determine a generalized entropy that can provide the Friedmann equations of any general gravity theory for all values of EOSs of the matter field. The further implications are discussed.