  Thermal behavior of effective, chiral condensate-dependent $U_A(1)$ anomaly
couplings is investigated using the functional renormalization group approach
in the $N_f = 3$ flavor meson model. We derive flow equations for anomaly
couplings that arise from instantons of higher topological charge, dependent
also on the chiral condensate. These flow equations are solved numerically for
the $|Q|=1,2$ topological sectors at finite temperature. Assuming that the
anomaly couplings at the ultraviolet scale may also exhibit explicit
temperature dependence, we calculate the thermal behavior of the effective
potential. In accordance with our earlier study, [G. Fejos and A. Patkos, Phys.
Rev. D{\bf 105}, 096007 (2022)], we find that for increasing temperatures, the
anomalous breaking of chiral symmetry tends to strengthen toward the
pseudocritical temperature ($T_C$) of chiral symmetry breaking. It is revealed
that below $T_C$, around $\sim$10\% of the $U_A(1)$ breaking arises from the
$|Q|=2$ topological sector. Correspondingly, a detailed analysis on the thermal
behavior of the mass spectrum is also presented.
