Is the $\bar \theta$ parameter of QCD constant?

Testing the cosmological variation of fundamental constants of Nature can provide valuable insights into new physics scenarios. While many such constraints have been derived for Standard Model coupling constants and masses, the $\bar\theta$ parameter of QCD has not been as extensively examined. 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 timescales. We find no evidence for a variation of $\bar\theta$ that could have implied an "iron-deficient" Universe at higher redshifts. By converting recent atomic clock constraints on a variation of constants, we infer $d({\bar\theta}^2)/dt \leq 6\times 10^{-15}\text{yr}^{-1}$, at 1-$\sigma$. Finally, we also sketch an axion model that 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.