  Thermal axion production in the early universe goes through several mass
thresholds, and the resulting rate may change dramatically across them.
Focusing on the KSVZ and DFSZ frameworks for the invisible QCD axion, we
perform a systematic analysis of thermal production across thresholds and
provide smooth results for the rate. The QCD phase transition is an obstacle
for both classes of models. For the hadronic KSVZ axion, we also deal with
production at temperatures around the mass of the heavy-colored fermion charged
under the Peccei-Quinn symmetry. Within the DFSZ framework, standard model
fermions are charged under this symmetry, and additional thresholds are the
heavy Higgs bosons masses and the electroweak phase transition. We investigate
the cosmological implications with a specific focus on axion dark radiation
quantified by an effective number of neutrino species and explore the discovery
reach of future CMB-S4 surveys.
