  We determine the first independent part of the g^6 coefficient in the weak
coupling expansion of the QCD pressure at high temperatures, the one
proportional to the maximal power of the number of quark flavors N_f. In
addition to introducing and developing computational methods that can be used
in evaluating other parts of the expansion, our calculation provides a result
that becomes dominant in the limit of large N_f and a fixed effective coupling
g_{eff}^2 = g^2 N_f/2.
