Cosmology, New Entropy and Thermodynamics of Apparent Horizon

We propose new nonadditive entropy of the apparent horizon $S_K=S_{BH}/(1+\gamma S_{BH}^2)$, where $S_{BH}$ is the Bekenstein--Hawking (BH) entropy and consider the description of new cosmology. When parameter $\gamma$ vanishes ($\gamma\rightarrow 0$) our entropy $S_K$ is converted into BH entropy $S_{BH}$. By using the holographic principle a new model of holographic dark energy is studied. We obtain the generalised Friedmann's equations for Friedmann--Lema\^{i}tre--Robertson--Walker (FLRW) spacetime for the barotropic matter fluid with equation of state $p=w\rho$. From the second modified Friedmann's equation we find a dynamical cosmological constant. The dark energy pressure $p_D$, density energy $\rho_D$ and the deceleration parameter $q$ corresponding to our model are computed. It is shown that at some EoS $w$ and parameter $\gamma$ there are phases of universe acceleration, deceleration and eternal inflation. Our model, with the help of the holographic principle, can describe the universe inflation and late time of the universe acceleration. We show the current deceleration parameter $q_0\approx -0.6$ is realized at some model parameters. The generalised entropy of the apparent horizon with the holographic dark energy model may be of interest for new cosmology.