case is also presented by the turning point method, and the correspondence between the Kerr–Rényi and the Kerr-AdS–Boltzmann approaches is discussed. In Sec. \[sec:summary\] we summarize our results and draw our conclusions. Throughout this paper we use units such as $c=G=\hbar=k_B=1$.

Rényi approach from nonadditive thermodynamics {#sec:Renyi}
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By replacing the additivity axiom to the weaker composability in the Shannon–Khinchin axiomatic definition of the entropy function, new type of entropy expressions arise. The composability axiom asserts, roughly speaking, that the entropy $S_{12}$ of a compound system consisting of two independent systems should be computable only in terms of the individual entropies $S_{1}$ and $S_{2}$. This means that there is a function $f(x,y)$ such that $$S_{12} = f (S_1, S_2),$$ for any independent systems. This property is of fundamental importance, since it implies that an entropic function is properly defined on macroscopic states of a given system, and it can be computed without having any information on the underlying microscopic dynamics. Composability is a key feature to ensure that the entropy function is physically meaningful. In a recent paper [@abe2001general], based on the concept of composability alone, Abe derived the most general functional form of those nonadditive entropy composition rules that 