  As the black hole entropy does not obey the area law on it. The area law,
given by $S\sim A^\delta\text{or}A^R$, where $\delta$ and $R$ , for
$0\leq\delta\leq 1$, $0<R\leq 1$, indicates the amount of quantum gravitational
deformation effects. This shows that gravity and thermodynamics are closely
related. Based on this deep connection, we tried to discuss the effects of
Sharma-Mittal entropy on the cosmological equations. We also investigated
whether the apparent horizon-enclosed universe was consistence with the
generalized second law of thermodynamics. Finally, using the gravity emergence
scenario, we derived the modified Friedmann equation and, compared it with that
derived from the first law of thermodynamics in the presence of Sharma-Mittal
entropy. It is observed that we can recover the results for standard cosmology
when $R,\delta=1$.
