  It has recently been shown that for certain classical non-dilatonic
$p$-branes, the entropy and temperature satisfy the ideal-gas relation $S\sim
T^{p}$ in the near-extremal regime. We extend these results to cases where the
dilaton is non-vanishing, but nevertheless remains finite on the horizon in the
extremal limit, showing that the ideal-gas relation is again satisfied. At the
classical level, however, this relation does break down if the dilaton diverges
on the horizon. We argue that such a divergence indicates the breakdown of the
validity of the classical approximation, and that by taking string and
worldsheet loop corrections into account, the validity of the
entropy/temperature relation may be extended to include these cases. This opens
up the possibility of giving a microscopic interpretation of the entropy for
all near-extremal $p$-branes.
