  We revisit the cosmological aspects of axion models. In the high-scale
inflation scenario, the Peccei-Quinn (PQ) symmetry is likely to be restored
during/after inflation. If the curvature of the PQ scalar potential at the
origin is smaller than its vacuum expectation value; for instance in a class of
SUSY axion models, thermal inflation happens before the radial component of the
PQ scalar (saxion) relaxes into the global minimum of the potential and the
decay of saxion coherent oscillation would produce too much axion dark
radiation. In this paper, we study how to avoid the overproduction of axion
dark radiation with some concrete examples. We show that, by taking account of
the finite-temperature dissipation effect appropriately, the overproduction
constraint can be relaxed since the PQ scalar can take part in the thermal
plasma again even after the PQ phase transition. We also show that it can be
further relaxed owing to the late time decay of another heavy CP-odd scalar, if
it is present.
