Pressure of hot QCD up tog6ln(1/g)

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, $g^6\ln \mathrm{}(1/g),$ 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 $g^6\ln \mathrm{}(1/g)\mathrm{}$ terms, once they are summed together with the so far unknown perturbative and nonperturbative $g^6$ terms, could potentially extend the applicability of the coupling constant series down to surprisingly low temperatures.