Hawking-R\'enyi black hole thermodynamics, Kiselev solution, and cosmic censorship

Explicit example, where the Hawking temperature of a black hole horizon is compatible with the black hole's R\'enyi entropy thermodynamic description, is constructed. It is shown that for every static, spherically symmetric, vacuum black hole space-time, a corresponding black hole solution can be derived, where the Hawking temperature is identical with the R\'enyi temperature, i.e. the one obtained from the R\'enyi entropy of the black hole via the 1st law of thermodynamics. In order to have this Hawking-R\'enyi type thermodynamic property, the black holes must be surrounded by an anisotropic fluid in the form of a Kiselev metric, where the properties of the fluid are uniquely determined by the mass of the black hole, $M$, and the R\'enyi parameter, {\lambda}. In the simplest Schwarzschild scenario, the system is found to be thermodynamically unstable, and the 3rd law of thermodynamics seems to play the role of a cosmic censor via placing an upper bound on the black hole's mass, by which preventing the black hole from loosing its horizon(s).