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

6+8 pages, 7 figures