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We strengthen the cosmological bound on the axion mass, by solving themomentum-dependent Boltzmann equations for axion-pion scatterings and by usinga phenomenological production rate derived from pion-pion scattering data,overcoming the breakdown of chiral perturbation theory. Using presentcosmological datasets we obtain $m_a\leq 0.24~\text{eV}$. To further improvethe bound and exploit the reach of upcoming cosmological surveys, reliablenon-perturbative calculations above the QCD crossover are needed.

Improved hot dark matter bound on the QCD axion