$$L(S_T)=\frac{1}{1-q}\ln\left[1+(1-q)S_T\right] \equiv S_R,$$ the Rényi expression (\[Srenyi\]) is reproduced, which, unlike the Tsallis formula, is additive for composition. In the limit of $q\rightarrow 1$ ($\lambda \rightarrow 0$), both the Tsallis- and the Rényi entropies recovers the standard Boltzmann-Gibbs description.

According to these results, in the present paper, in order to describe the non-Boltzamannian nature of Kerr black holes, we consider the Tsallis statistics as the simplest, nonadditive, parametric but equilibrium compatible extension of the Boltzamann-Gibbs theory which also satisfies Abe’s formula. On the other hand, in order to satisfy the zeroth law of thermodynamics, we follow the formal logarithm method of Biró and Ván, and rather the Tsallis description, we consider the Rényi entropy for the thermodynamics of the problem. Since the Rényi definition is additive, it satisfies all laws of thermodynamics, and compared to the Boltzmann picture, it has the advantage of having a free parameter which can be accounted to describe the effects of nonlocality in our approach. The thermodynamics of Schwarzschild black holes in this model has been studied in [@Biro:2013cra], and the corresponding stability problem has been investigated in [@Czinner:2015eyk].

Kerr black holes {#sec:Kerr}
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The spacetime metric that describes the geometry of a rotating black 