  We analyze the contribution of the $\eta'(958)$ meson in the first two
non-trivial moments of the QCD topological charge distribution, namely, the
topological susceptibility and the fourth-order cumulant of the vacuum energy
density. We perform our study within U(3) Chiral Perturbation Theory up to
next-to-next-to-leading order in the combined chiral and large-$N_c$ expansion.
We also describe the temperature dependence of these two quantities and compare
them with previous analyses in the literature. In particular, we discuss the
validity of the thermal scaling of the topological susceptibility with the
quark condensate, which is intimately connected with a Ward Identity relating
both quantities. We also consider isospin breaking corrections from the vacuum
misalignment at leading order in the U(3) framework.
