New entropy, thermodynamics of apparent horizon and cosmology

Here, we consider new nonadditive entropy of the apparent horizon $S_K=S_{BH}\mathrm{/}(1+\gamma S_{BH})$ with $S_{BH}$ being the Bekenstein–Hawking entropy. This is an alternative of the Rényi and Tsallis entropies, that allows us, by utilising the holographic principle, to develop entropic (holographic) dark energy. When $\gamma \to 0$ our entropy becomes the Bekenstein–Hawking entropy $S_{BH}$. The generalized Friedmann equations for Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime for the barotropic matter fluid with $p=w\rho$ were obtained. We compute the dark energy pressure $p_D$, density of energy ${\rho }_D$, the normalized density parameters ${\mathrm{\Omega }}_D$, ${\mathrm{\Omega }}_m$ and the deceleration parameter $q$ of the universe corresponding to our model. From the second modified Friedmann equation a dynamical cosmological constant was obtained. We show that at some model parameters $w$ and $\gamma$ we obtain ${\mathrm{\Omega }}_{m0}\approx 0.315$ and $q_0\approx -0.535$ which are in agreement with the Planck data. \cite{Aghanim}. It was shown that the model under consideration possesses the phantom divide for the EoS of dark energy. Thus, our model, by virtue of the holographic principle, can describe the universe inflation and the late time of the universe acceleration. It is shown that entropic cosmology with our entropy proposed is equivalent to cosmology based on the teleparallel gravity with the function $F(T)$ obtained. The holographic dark energy model with the generalised entropy of the apparent horizon can be of interest for new cosmology.