For Schwarzschild black holes, generalized non-extensive entropies such as Rényi, Tsallis, and four- and five-parameter forms cannot reproduce both the Hawking temperature and the ADM mass; only the Bekenstein-Hawking entropy satisfies both.
Modified gravity theories from the Barrow hypothesis
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Barrow proposed that quantum gravity effects might introduce fractal corrections to the area of the event horizon of black holes. The area law gets modified as $S \propto A^{1+\Delta/2}$, with $0\leq\Delta\leq 1$. It was so far unclear whether this assumption could lead to meaningful quantum gravity theories beyond general relativity. In this paper, we argue that this is indeed the case. In particular, assuming $\Delta$ to be a radial function, we show that the Barrow hypothesis, together with the Jacobson's approach can generate non-trivial modified gravity theories.
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Black Holes Thermodynamics and Generalised Non-Extensive Entropy
For Schwarzschild black holes, generalized non-extensive entropies such as Rényi, Tsallis, and four- and five-parameter forms cannot reproduce both the Hawking temperature and the ADM mass; only the Bekenstein-Hawking entropy satisfies both.