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We compute the free energy density $F$ for gauge theories, with fermions, athigh temperature and zero chemical potential. In the expansion $F=T^4 [c_0+c_2g^2+c_3 g^3+(c'_4\ln g+c_4)g^4+ (c'_5\ln g+c_5)g^5+O(g^6)]$, 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 thenon-Abelian case the last power of $g$ that can be computed within perturbationtheory. We find that the $g^5$ term receives no contributions from overlappingdouble-frequency sums and that $c'_5$ vanishes.Comment: 31 pages, 6 figures, LaTeX; minor beautifications, reference list  extended, version to be published in Phys.Rev.

physical review. d. particles, fields, gravitation, and cosmology/physical review. d. particles and fields

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