  Light axions can potentially leave a cosmic background, just like neutrinos.
We complete the study of thermal axion production across the electroweak scale
by providing a smooth and continuous treatment through the two phases. Focusing
on both flavor conserving and violating couplings to third generation quarks,
we compute the amount of axions produced via scatterings and decays of thermal
bath particles. We perform a model independent analysis in terms of axion
effective couplings, and we also make predictions for specific microscopic QCD
axion scenarios. This observable effect, parameterized as it is conventional by
an effective number of additional neutrinos, is above the $1\sigma$ sensitivity
of future CMB-S4 surveys. Moreover, if one assumes no large hierarchies among
dimensionless axion couplings to standard model particles, future axion
helioscopes will provide a complementary probe for the parameter region we
study.
