The measurement of γ-rays originating from active galactic nuclei offers the unique opportunity to study the propagation of very-high-energy photons over cosmological distances. Most prominently, γ-rays interact with the extragalactic background light (EBL) to produce e+e− pairs, imprinting an attenuation signature on γ-ray spectra. The e$^+$e$^−$ pairs can also induce electromag- netic cascades whose detectability in γ-rays depends on the intergalactic magnetic field (IGMF). Furthermore, physics beyond the Standard Model such as Lorentz invariance violation (LIV) or oscillations between photons and weakly interacting sub-eV particles (WISPs) could affect the propagation of γ-rays. The future Cherenkov Telescope Array (CTA), with its unprecedented γ- ray source sensitivity, as well as enhanced energy and spatial resolution at very high energies, is perfectly suited to study cosmological effects on γ-ray propagation. Here, we present first results of a study designed to realistically assess the capabilities of CTA to probe the EBL, IGMF, LIV, and WISPs.