Modified Cosmology from Mass-to-Horizon Relation: Observational Bounds

We constrain the class of modified cosmologies derived in Paper I [1] from a generalized mass-to-horizon relation (MHR) that enforces thermodynamic consistency between the Cai-Kim horizon temperature and generalized horizon entropies. The modified Friedmann equations depend on an entropy exponent $m$, an MHR coupling parameter $\gamma$, and an entanglement-correction amplitude $f_B$, with standard $\Lambda$CDM recovered in the appropriate limit. Using Pantheon$+$/SH0ES Type~Ia supernovae, cosmic chronometers, DESI DR2 baryon acoustic oscillations, and Planck 2018 CMB distance priors, we constrain eight physically motivated sub-cases via Markov chain Monte Carlo and compare models through the Bayesian log-evidence. The entropy exponent is tightly bounded, $|m-1|\lesssim O(10^{-4})$ when the MHR coupling is fixed ($\gamma=1$), relaxing to $O(10^{-3})$ along the $m$-$\gamma$ degeneracy, excluding any macroscopically significant departure from standard horizon thermodynamics. Freeing the MHR coupling parameter or the entanglement amplitude raises the inferred Hubble constant to $h\simeq0.70$-$0.71$, reducing the CMB-SH0ES tension from ${\sim}4\sigma$ to ${\sim}1.2$-$2.6\sigma$, but no scenario fully resolves it within a flat universe. The Bayesian log-evidence nevertheless disfavors every extension relative to $\Lambda$CDM in all dataset combinations ($-16\lesssim\Delta\ln Z \lesssim-1$): the improved fits obtained when the SH0ES calibration is included reflect an absorption of the Hubble tension by the additional parameters rather than genuine evidence for modified horizon entropy.