  We study the effects of failure to equilibrate the squared topological charge
$Q^2$ on lattice calculations of pseudoscalar masses and decay constants. The
analysis is based on chiral perturbation theory calculations of the dependence
of these quantities on the QCD vacuum angle $\theta$. For the light-light
partially quenched case, we rederive the known chiral perturbation theory
results of Aoki and Fukaya, but using the nonperturbatively-valid chiral theory
worked out by Golterman, Sharpe and Singleton, and by Sharpe and Shoresh. We
then extend these calculations to heavy-light mesons. Results when staggered
taste-violations are important are also presented. The derived $Q^2$ dependence
is compared to that of simulations using the MILC collaboration's ensembles of
lattices with four flavors of HISQ dynamical quarks. We find agreement, albeit
with large statistical errors. These results can be used to correct for the
leading effects of unequilibrated $Q^2$, or to make estimates of the systematic
error coming from the failure to equilibrate $Q^2$. In an appendix, we show
that the partially quenched chiral theory may be extended beyond a lower bound
on valence masses discovered by Sharpe and Shoresh. Subtleties occurring when a
sea-quark mass vanishes are discussed in another appendix.
