  We determine the 2-loop effective gauge coupling of QCD at high temperatures,
defined as a matching coefficient appearing in the dimensionally reduced
effective field theory. The result allows to improve on one of the classic
non-perturbative probes for the convergence of the weak-coupling expansion at
high temperatures, the comparison of full and effective theory determinations
of an observable called the spatial string tension. We find surprisingly good
agreement almost down to the critical temperature of the deconfinement phase
transition. We also determine one new contribution of order O(g^6T^4) to the
pressure of hot QCD.
