are compatible with homogeneous equilibrium. Assuming that $f(S_1, S_2)$ is a $C^2$ class symmetric function, Abe showed that the most general, equilibrium compatible composition rule takes the form $$\label{eq:Abe}
H_{\lambda}(S_{12})=H_{\lambda}(S_1)+H_{\lambda}(S_2)+\lambda H_{\lambda}(S_1)H_{\lambda}(S_2),$$ where $H_{\lambda}$ is a differentiable function of $S$ and $\lambda\in\mathbb{R}$ is a constant parameter. Later on, this result has been extended to non-homogeneous systems as well [@PhysRevE.83.061147], where not only the entropy, but the energy function is also considered to be nonadditive.

The simplest and perhaps the most well-known nonadditive entropy composition rule can be obtained from (\[eq:Abe\]) by setting $H_{\lambda}(S)$ to be the identity function, i.e. $H_{\lambda}(S)=S$. In this case Abe’s equation becomes $$\label{eq:tsallis_composition}
S_{12}=S_1+S_2+\lambda S_1 S_2,$$ which results the familiar Tsallis composition rule with $\lambda=1-q$ [@Tsallis:1987eu], and the corresponding entropy definition is given in (\[TsS\]). The nonextensive Tsallis statistics is widely investigated in many research fields from natural to social sciences, an updated bibliography on the topic can be found in [@Tsallis-bib]. This approach has also been studied in the problem of black hole thermodynamics (see e.g. [@Tsallis:2012js] and references therein), and our starting point in considering a more general entropy definition for black holes than the one based on the Boltzamann-Gibbs statistics is also the Tsallis 