  Hot axions, thermally produced in the Early Universe, would contribute to
dark radiation and are thus subject to present and future constraints from
$N_{\rm eff}$. In this paper we quantify the contribution to $N_{\rm eff}$ and
its uncertainty in models with axion-gluon couplings from thermal dynamics
above the QCD transition. In more detail, we determine the leading-order
thermal axion production rate for axion momenta of the order of the temperature
adopting three different schemes for the incorporation of the collective
dynamics of soft gluons. We show how these three schemes extrapolate
differently into the regime of softer axion production, thus giving us a
quantitative handle on the theory uncertainty of the rate. Upon solving the
Boltzmann equation, we find that this theory uncertainty translates to an
uncertainty of at most 0.002 for $N_{\rm eff}$. The uncertainty from common
momentum-averaged approximations to the Boltzmann equation is smaller. We also
comment on existing rate determinations in the literature and discuss how QCD
transition dynamics would need to be integrated into our results.
