  We show that the commonly adopted hot dark matter (HDM) bound on the axion
mass $m_a \lesssim$ 1 eV is not reliable, since it is obtained by extrapolating
the chiral expansion in a region where the effective field theory breaks down.
This is explicitly shown via the calculation of the axion-pion thermalization
rate at the next-to-leading order in chiral perturbation theory. We finally
advocate a strategy for a sound extraction of the axion HDM bound via lattice
QCD techniques.
