Non-Extensive Black Hole Thermodynamics Estimate for Power-Law Particle Spectra

We point out that by considering the Hawking-Bekenstein entropy of Schwarzschild black hole horizons as a non-extensive Tsallis entropy, its additive formal logarithm, coinciding with the Renyi entropy, generates an equation of state with positive heat capacity above a threshold energy. Based on this, the edge of stability is conjectured to be trans-Planckian, i.e. being in the quantum range. From this conjecture an estimate arises for the q-parameter in the Renyi entropy, (q=2/pi^2), also manifested in the canonical power-law distribution of high energy particles (q ~ 1.2 for quark matter).