Non-extensive and quasi-homogeneous geometrothermodynamics

We study the thermodynamic properties of black holes, taking into account the non-extensive character of their entropy at the thermodynamic and statistical level. To this end, we assume that the R\'enyi entropy determines the fundamental thermodynamic equation of black holes and is represented by a quasi-homogeneous function. As a consequence, the R\'enyi parameter turns out to be an independent thermodynamic variable, which must be treated in the framework of extended thermodynamics. As a particular example, we use the formalism of geometrothermodynamics to show that the Schwarzschild black hole can become stable for certain values of the R\'enyi parameter.