The Cherenkov Telescope Array (CTA), the new-generation ground-based observatory for $gamma$-ray astronomy, provides unique capabilities to address significant open questions in astrophysics, cosmology, and fundamental physics. We study some of the salient areas of $gamma$-ray cosmology that can be explored as part of the Key Science Projects of CTA, through simulated observations of active galactic nuclei (AGN) and of their relativistic jets. Observations of AGN with CTA will enable a measurement of $gamma$-ray absorption on the extragalactic background light with a statistical uncertainty below 15% up to a redshift $z=2$ and to constrain or detect $gamma$-ray halos up to intergalactic-magnetic-field strengths of at least 0.3pG. Extragalactic observations with CTA also show promising potential to probe physics beyond the Standard Model. The best limits on Lorentz invariance violation from $gamma$-ray astronomy will be improved by a factor of at least two to three. CTA will also probe the parameter space in which axion-like particles could constitute a significant fraction, if not all, of dark matter. We conclude on the synergies between CTA and other upcoming facilities that will foster the growth of $gamma$-ray cosmology.
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