  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).
