Paper III — Background Cosmology in the Time-First Formulation This paper develops the cosmological sector of the time-first formulation of gravity, where a single temporal potential Φ defines the lapse (and thus physical clock rates), while spatial geometry follows from the ADM constraints and evolution equations. We prove exact equivalence to standard FRW background cosmology and provide a complete, self-contained “dictionary” between FRW variables and the time-first variables: $a = e^{-\Phi}$, $H = -\,d\Phi/d\tau$, with $d\tau = e^{\Phi} dt$. On this foundation we build two minimal tracks for late-time acceleration: (i) a Λ track statistically indistinguishable from ΛCDM at background level, and (ii) a phenomenological scalar-potential track $V(\Phi)$ that preserves GR’s tensor content (two propagating polarizations) while offering a transparent parameterization for data fitting. The paper restricts strictly to background (zero-mode) dynamics; perturbations and laboratory-scale quantum predictions are deferred to the companion papers. We implement a conservative likelihood using Type Ia supernovae, BOSS DR12 BAO, and a CMB distance prior to show that time-first cosmology is statistically equivalent to ΛCDM at background level. The mapping makes several aspects of cosmology especially transparent: (1) initial-condition language in terms of Φ (Big-Bang limit $a\to0 \Leftrightarrow \Phi\to+\infty$); (2) a clear separation between the geometric lapse Φ (non-propagating) and any matter scalar used for dark energy; and (3) compact evolution equations that streamline numerical work. The result is a tightly constrained, experimentally anchored reformulation that unifies background expansion and observable distance measures within a single scalar geometric degree of freedom—while remaining classically equivalent to GR. Scope: Background cosmology and data fits (no perturbations).Companion works: Paper I (Foundations and classical equivalence), Paper II (Experimental/quantum signatures), Paper IV (Perturbations and structure).Reproducibility: All equations and priors are provided; data sets are standard community releases (SNe, BAO, CMB distance prior).Status: Accepted to viXra (ID 2508.0048) (August 2025). This Zenodo record will serve as the canonical DOI; future revisions may be posted here (Zenodo concept DOI versioning).