Thermodynamics of black holes with Rényi entropy from classical gravity

The nonextensive nature of black holes is one of the most intriguing discoveries. In fact, the black hole entropy is a nonextensive quantity that scales by its surface area at the event horizon. In our work, we extend the thermodynamic phase space of black holes by treating the nonextensive parameter analyzed via the Rényi entropy as the thermodynamic variable. Using Euler's theorem for a homogeneous function of the black holes' mass, the compatible Smarr formula and the first law of black hole thermodynamics can be obtained. It is also demonstrated that, by keeping the same form of the black hole mass, the Rényi temperature is straightforwardly defined as proposed in the literature. Since many different types of black holes can indeed be successfully treated with such a procedure, our consideration is fairly general. It is worthwhile to argue that the black hole thermodynamics in Rényi statistics is rooted from the relation among geometric quantities in the same way as the standard approach corresponding to the Gibbs–Boltzmann statistics. Even though our results are based on classical gravity, they may pave the way to derive the Rényi temperature using the notion of quantum field in curved spacetime.

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