  This is the last in a series of papers on the topological susceptibility in
the interacting instanton liquid model (IILM). We will derive improved finite
temperature interactions to study the thermodynamic limit of grand canonical
Monte Carlo simulations in the quenched and unquenched case with light,
physical quark masses. In particular, we will be interested in chiral symmetry
breaking. The paper culminates by giving, for the first time, a well-motivated
temperature-dependent axion mass. Especially, this work finally provides a
computation of the axion mass in the low temperature regime, $m^2_a f^2_a =
1.46 10^{-3}\Lambda^4 \frac{1+0.50 T/\Lambda}{1+(3.53 T/\Lambda)^{7.48}}$. It
connects smoothly to the high temperature dilute gas approximation; the latter
is improved by including quark threshold effects. To compare with earlier
studies, we also provide the usual power-law $m^2_a = \frac{\alpha_a
\Lambda^4}{f_a^2 (T/\Lambda)^n}$, where $\Lambda=400\units{MeV}$, $n=6.68$ and
$\alpha=1.68 10^{-7}$.
