  We show that the Virasoro Gibbs state accurately describes the thermodynamics
of BTZ black holes at large temperatures and in the strong-coupling regime
$c<1$. We first give a simple heuristic argument by showing that at high
temperatures and arbitrary $c$, the energies are much larger than Planck mass.
Then we give a detailed analysis of the quantum fluctuations of the Gibbs state
on unitary irreducible representations of Virasoro group and explicitly show
that they go to zero in the high-temperature limit by means of representation
theory results. This implies the state has a sensible thermodynamic limit which
actually holds for any $c$. Finally, the matching with BTZ thermodynamics for
$c<1$ is obtained by using the known asymptotic behaviour of the characters of
the Virasoro discrete series. This result supports the idea that minimal models
could describe quantum gravity at strong coupling. We make no use of the
Euclidean path integral nor assume modular invariance of the partition
functions, although we comment on them along the way.
