  The free energy density, or pressure, of QCD has at high temperatures an
expansion in the coupling constant g, known so far up to order g^5. We compute
here the last contribution which can be determined perturbatively, g^6 ln(1/g),
by summing together results for the 4-loop vacuum energy densities of two
different three-dimensional effective field theories. We also demonstrate that
the inclusion of the new perturbative g^6 ln(1/g) terms, once they are summed
together with the so far unknown perturbative and non-perturbative g^6 terms,
could potentially extend the applicability of the coupling constant series down
to surprisingly low temperatures.
