  If the QCD axion is a significant component of dark matter, and if the
universe was once hotter than a few hundred MeV, the axion relic abundance
depends on the function $\chi(T)$, the temperature-dependent topological
susceptibility. Uncertainties in this quantity induce uncertainties in the
axion mass as a function of the relic density, or vice versa. At high
temperatures, theoretical uncertainties enter through the dilute instanton gas
computation, while in the intermediate and strong coupling regime, only lattice
QCD can determine $\chi(T)$ precisely. We reassess the uncertainty on the
instanton contribution, arguing that it amounts to less than $20\%$ in the
effective action, or a factor of 20 in $\chi$ at $T=1.5$ GeV. We then combine
the instanton uncertainty with a range of models for $\chi(T)$ at intermediate
temperatures and determine the impact on the axion relic density. We find that
for a given relic density and initial misalignment angle, the combined
uncertainty amounts to a factor of 2-3 in the zero-temperature axion mass.
