  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.
