  We show that the upper bound of the classical QCD axion window can be
significantly relaxed for low-scale inflation. If the Gibbons-Hawking
temperature during inflation is lower than the QCD scale, the initial QCD axion
misalignment angle follows the Bunch-Davies distribution. The distribution is
peaked at the strong CP conserving minimum if there is no other light degree of
freedom contributing to the strong CP phase. As a result, the axion
overproduction problem is significantly relaxed even for an axion decay
constant larger than $10^{12}$GeV. We also provide concrete hilltop inflation
models where the Hubble parameter during inflation is comparable to or much
smaller than the QCD scale, with successful reheating taking place via
perturbative decays or dissipation processes.
