  We compute the free energy density $F$ for gauge theories, with fermions, at
high temperature and zero chemical potential. In the expansion $F=T^4 [c_0+c_2
g^2+c_3 g^3+(c'_4\ln g+c_4)g^4+ (c'_5\ln g+c_5)g^5+O(g^6)]$, we determine
$c'_5$ and $c_5$ analytically by calculating two- and three-loop diagrams. The
$g^5$ term constitutes the first correction to the $g^3$ term and is for the
non-Abelian case the last power of $g$ that can be computed within perturbation
theory. We find that the $g^5$ term receives no contributions from overlapping
double-frequency sums and that $c'_5$ vanishes.
