Entropic Cosmology with New Entropy

We study entropic cosmology with new nonadditive entropy of the apparent horizon $S_K=S_{BH}/(1+\sigma S_{BH})^2$ where $S_{BH}$ is the Bekenstein--Hawking entropy. Making use of the thermodynamics-gravity correspondence a new model of dark energy is analyzed. We obtain the generalised Friedmann equations for Friedmann--Lema\^{i}tre--Robertson--Walker spatially flat universe with the barotropic matter fluid possessing the equation of state (EoS) $w=p/\rho$. The dark energy pressure $p_D$, the energy density $\rho_D$, and the deceleration parameter $q$ of the universe are computed. At some model parameters $w$ and $\sigma$ we obtain the normalized density parameter of the matter $\Omega_{m}\approx 0.315$ and the deceleration parameter $q_0\approx -0.535$ for the current epoch that are in agreement with the Planck data. We show that our model has the phantom divide for the EoS of dark energy. The model under consideration, with the help of the thermodynamics-gravity correspondence, describes the universe inflation and the late time of the universe acceleration. It is proven that the suggested entropic cosmology being equivalent to cosmology within the teleparallel gravity with the function $F(T)$ which we computed.