  We study cosmological effects of explicit Peccei-Quinn breaking on the QCD
axion dark matter. We find that the axion abundance decreases or increases
significantly depending on the initial position, even for a tiny Peccei-Quinn
breaking that satisfies the experimental bound of the neutron electric dipole
measurements. If the axion first starts to oscillate around a wrong vacuum and
if it gets trapped there until the false vacuum disappears due to
non-perturbative QCD effects, its abundance increases significantly and is
independent of the decay constant $f_a$, as first pointed out in [JHEP 06
(2016) 150]. Thus, the axion produced by the trapping mechanism can explain
dark matter even when the decay constant is close to the lower limit due to
stellar cooling arguments. On the other hand, if the axion starts to oscillate
about a potential minimum close to the low-energy vacuum, its abundance is
significantly reduced because of the adiabatic suppression mechanism. This
relaxes the upper limit of the axion window to large values of $f_a$. We also
discuss how the axionic isocurvature perturbation is affected by the
Peccei-Quinn breaking term, and show that it can be suppressed in both regimes.
In particular, the isocurvature bound on the inflation scale is relaxed by many
orders of magnitudes for $f_a \lesssim 10^{11}{\rm GeV}$ compared to the
conventional scenario.
