  In this work we compute the axion mass and, from this (exploiting a
well-known relation), we also derive an expression for the QCD topological
susceptibility in the finite-temperature case, both below and above the chiral
phase transition at $T_c$, making use of a chiral effective Lagrangian model
which includes the axion, the scalar and pseudoscalar mesons and implements the
$U(1)$ axial anomaly of the fundamental theory. We also provide a numerical
estimate of the topological susceptibility at $T=0$ (in the physical case of
three light quark flavors) and discuss the question of the temperature and
quark-mass dependence of the topological susceptibility in the high-temperature
regime.
