  We study the parameter space of cold dark matter axions in two cosmological
scenarios with non-standard thermal histories before Big Bang nucleosynthesis:
the Low Temperature Reheating (LTR) cosmology and the kination cosmology. If
the Peccei-Quinn symmetry breaks during inflation, we find more allowed
parameter space in the LTR cosmology than in the standard cosmology and less in
the kination cosmology. On the contrary, if the Peccei-Quinn symmetry breaks
after inflation, the Peccei-Quinn scale is orders of magnitude higher than
standard in the LTR cosmology and lower in the kination cosmology. We show that
the axion velocity dispersion may be used to distinguish some of these
non-standard cosmologies. Thus, axion cold dark matter may be a good probe of
the history of the Universe before Big Bang nucleosynthesis.
