  We recently obtained an estimate of the axion mass based on the hypothesis
that axions make up most of the dark matter in the universe. A key ingredient
for this calculation was the temperature-dependence of the topological
susceptibility of full QCD. Here we summarize the calculation of the
susceptibility in a range of temperatures from well below the finite
temperature cross-over to around 2 GeV. The two main difficulties of the
calculation are the unexpectedly slow convergence of the susceptibility to its
continuum limit and the poor sampling of nonzero topological sectors at high
temperature. We discuss how these problems can be solved by two new techniques,
the first one with reweighting using the quark zero modes and the second one
with the integration method.
