Correction to the Friedmann equation with Sharma-Mittal entropy: A new perspective on cosmology

Abstract: 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 $R,\delta=1$.