The big bang quantum cosmology model introduces the trace $J$ of the Schouten tensor as a form of dynamic dark energy. Together with cold dark matter (CDM), these components form the so-called $J\mathrm{CDM}$ cosmology model, proposed by van Putten [J. High Energy Astrophys. 45, 194 (2025)], which offers a potential resolution to the Hubble tension. We derive the constraints on the $J\mathrm{CDM}$ cosmology model, utilizing early- and late-time cosmological data including cosmic microwave background, baryon acoustic oscillations released by the Dark Energy Spectroscopic Instrument (DESI), cosmic chronometers, and type Ia supernovae. For a flat universe, the $J\mathrm{CDM}$ model yields $H_0=66.95\pm 0.51\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and ${\mathrm{\Omega }}_m=0.3419\pm 0.0065$, results that are consistent with early-Universe observations but exhibit a higher ${\mathrm{\Omega }}_m$ compared to the $\mathrm{\Lambda }\mathrm{CDM}$ model. In the case of a nonflat universe, $J\mathrm{CDM}$ favors a slightly curved geometry with ${\mathrm{\Omega }}_k=0.0154\pm 0.0027$, leading to $H_0=69.13\pm 0.56\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and ${\mathrm{\Omega }}_m=0.3477\pm 0.0074$. The increase in $H_0$ in the nonflat scenario suggests a geometric degeneracy between spatial curvature and $H_0$. We also investigate the internal inconsistencies present in DESI data and evaluate their impacts on cosmological parameter constraints. Our analysis shows that while the $J\mathrm{CDM}$ model, which is constructed from first principles without free parameters beyond those of $\mathrm{\Lambda }\mathrm{CDM}$, agrees excellently with late-time cosmology, it struggles to simultaneously match early-Universe observations in a fully self-consistent manner.

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• Cosmology