Modified cosmology from the thermodynamics of apparent horizon

In the realm of the Bekenstein-Hawking entropy, the thermodynamics of apparent horizon bridges with the usual FLRW (Friedmann-Lema\^{i}tre-Robertson-Walker) equation only for a special case where the matter field is given by a perfect fluid having equation of state (EoS) parameter $= -1$, i.e $p = -\rho$ with $\rho$ and $p$ represent the energy density and the pressure of the fluid, respectively. To include the case $p \neq -\rho$, we consider the modification of the Bekenstein-Hawking entropy in the present work. In particular, we develop an entropy function that leads to the usual FLRW equations, for a $general$ EoS of the matter fluid given by $p = w\rho$, directly from the thermodynamics of the apparent horizon. The newly developed entropy acquires a correction over the Bekenstein-Hawking entropy and differs from the known entropies like the Tsallis, R\'{e}nyi, Barrow, Sharma-Mittal, Kaniadakis, and Loop Quantum Gravity entropies proposed so far. Based on this finding, we examine how the Friedmann equations of the apparent horizon cosmology are accordingly modified if one starts with a general entropy depending on the Bekenstein-Hawking entropy. This results in some interesting cosmological consequences during the early and late stages of the universe.