How fundamental is entropy? From non-extensive statistics and black hole physics to the holographic dark universe

Abstract: We propose a new entropy construct that generalizes the Tsallis, R\'enyi, Sharma-Mittal, Barrow, Kaniadakis, and Loop Quantum Gravity entropies and reduces to the Bekenstein-Hawking entropy in a certain limit. This proposal is applied to the Schwarzschild black hole and to spatially homogeneous and isotropic cosmology, where it is shown that it can potentially describe inflation and/or holographic dark energy.