Is the $\bar \theta$ parameter of QCD constant?
  Testing the variation of fundamental constants of Nature can provide valuable
insights into new physics scenarios. While many constraints have been derived
for Standard Model coupling constants and masses, the $\bar{\theta}$ parameter
of QCD has often been ignored in these studies. This letter discusses
potentially promising paths to investigate the time dependence of the
$\bar{\theta}$ parameter. While laboratory searches for CP-violating signals of
$\bar{\theta}$ yield the most robust bounds on today's value of $\bar{\theta}$,
we show that CP-conserving effects provide constraints on the variation of
$\bar{\theta}$ over cosmological time scales. We find no significant evidence
for a variation of $\bar{\theta}$ except for a mild hint around $z\sim 4$ which
would imply an "iron-deficient" Universe at high redshifts. Finally, we also
sketch an axion model which results in a varying $\bar{\theta}$ and could lead
to excess diffuse gamma ray background, from decays of axions produced in high
redshift supernova explosions.
