  In this paper we study the proposal present in \cite{1} concerning the
statistical description of trapped gravitons and applied to derive the
semi-classical black hole (BH) entropy $S_{BH}$. We study the possible
configurations depending on physically reasonable expressions for the internal
energy $U$. In particular, we show that expressions for $U\sim R^k, k\geq 1$,
with $R$ the radius of the confining spherical box, can have a semi-classical
description, while behaviors with $k<1$ derive from thermodynamic or quantum
fluctuations. There, by taking a suitable physically motivated expression for
$U(R)$, we obtain the well known logarithmic corrections to the BH entropy,
with the usual behaviors present in the literature of BH entropy. Moreover, a
phase transition emerges with a positive specific heat $C$ at Planckian lengths
instead of the usual negative oneat non-Planckian scales, in agreement with
results present in the literature. Finally, we show that evaporation stops at a
radius $R$ of the order of the Planck length.
