The history of photon production and galaxy evolution since the epoch of reionization is encoded in the extragalactic background light (EBL). Above an energy threshold, $\gamma$-rays can interact with the optical and infrared photons that dominate the EBL, resulting in an absorption imprint in the spectra of extragalactic sources. The combined observations of the current generation of ground-based $\gamma$-ray instruments have recently enabled the first purely parametric $\gamma$-ray measurement of the EBL spectrum at $z=0$ that is independent of models of the evolution of the EBL with redshift. In this work, we extend this $\gamma$-ray cosmology analysis to the next generation of $\gamma$-ray observatories, the Cherenkov Telescope Array Observatory (CTAO), which will bring improved sensitivity and energy resolution, and broader energy range. Using simulations of almost 3000 hours of observations, we demonstrate the unprecedented precision across the EBL spectrum that CTAO could achieve, and we explore the implications of such a precision for $\gamma$-ray cosmology. We show the potential for CTAO to measure $H_0$, the expansion rate of the Universe at $z=0$, as well as to place constraints on diffuse emissions from exotic processes.