  The free energy of the maximally supersymmetric SU(N) gauge theory at
temperature T is expected to scale, in the large N limit, as N^2 T^4 times a
function of the 't Hooft coupling, f(g_{YM}^2 N). In the strong coupling limit
the free energy has been deduced from the near-extremal 3-brane geometry, and
its normalization has turned out to be 3/4 times that found in the weak
coupling limit. In this paper we calculate the leading correction to this
result in inverse powers of the coupling, which originates from the R^4 terms
in the tree level effective action of type IIB string theory. The correction to
3/4 is positive and of order (g_{YM}^2 N)^{-3/2}. Thus, f(g_{YM}^2 N) increases
as the 't Hooft coupling is decreased, in accordance with the expectation that
it should be approaching 1 in the weak coupling limit. We also discuss similar
corrections for other conformal theories describing coincident branes. In
particular, we suggest that the coupling-independence of the near extremal
entropy for D1-branes bound to D5-branes is related to the vanishing of the
Weyl tensor of AdS_3\times S^3.
