Not yet reviewed by Pith; the record is open.

Scattering and decay processes of thermal bath particles involving heavy leptons can dump hot axions in the primordial plasma around the QCD phase transition. We compute their relic density, parameterized by an effective number $\Delta N_{\rm eff}$ of additional neutrinos. For couplings allowed by current bounds, production via scattering yields $\Delta N_{\rm eff} \lesssim 0.6$ and $\Delta N_{\rm eff} \lesssim 0.2$ for the cases of muon and tau, respectively. Flavor violating tau decays to a lighter lepton plus an axion give $\Delta N_{\rm eff} \lesssim 0.3$. Such values of $\Delta N_{\rm eff}$ can alleviate the tension between the direct local measurement of the Hubble constant $H_0$ and the inferred value from observations of the Cosmic Microwave Background, assuming $\Lambda$CDM. We analyze present cosmological data from the Planck collaboration and baryon acoustic oscillations with priors given in terms of the axion-lepton couplings. For axions coupled to muons, the tension can be alleviated below the 3$\sigma$ level. Future experiments will measure $\Delta N_{\rm eff}$ with higher precision, providing an axion discovery channel and probing the role of hot axions in the $H_0$ tension.