  We study the QCD topology and axion properties at finite temperature and
chemical potential in the framework of the two-flavor Nambu$-$Jona-Lasinio
model. We find that the behaviors of the two lowest cumulants of the QCD
topological charge distribution and axion properties are highly sensitive to
the critical behavior of the chiral phase transition. In particular, the
topological susceptibility and the axion mass follow the response of the chiral
condensate to temperature and chemical potential, showing that both quantities
decrease monotonically with the increment of temperature and/or chemical
potential. However, it is important to note that the normalized fourth cumulant
behaves differently depending on the temperature. At low temperatures, it is a
non-monotonic function of the chemical potential, while at high temperatures,
it monotonically decreases. Additionally, its value invariably approaches the
asymptotic value of $b_2^{\text {inst }}=-1/12$, predicted by the dilute
instanton gas model. We also observe that with the increase in chemical
potential at relatively low temperatures, the axion self-coupling constant
exhibits a sharp peak around the critical point, which can even be more than
twice its vacuum value. After that, the self-coupling drops sharply to a much
lower value than its vacuum value, eventually approaching zero in the high
chemical potential limit. The finding that the axion self-coupling constant is
significantly enhanced in high-density environments near the chiral phase
transition could lead to the creation or enhancement of an axion Bose-Einstein
condensate in compact astrophysical objects.
