Holographic and Gravity-Thermodynamic Approaches in Entropic Cosmology: Bayesian Assessment using late-time Data

We investigate the cosmological implications of entropy-based approaches in the context of Holographic Dark Energy (HDE) and Gravity-Thermodynamics (GT) formalisms. We utilise the extended Barrow entropy form, with the index parameter $\Delta$, representing the fractal dimension of the horizon. We also test implementing different parameter ranges for $\Delta$, which can be extended to Tsallis' interpretation within the same formal cosmology. We perform a Bayesian analysis to constrain the cosmological parameters using the Pantheon+, more recent DESy5, DESI, and, as a supplement, Quasar datasets. We find that the HDE model within almost all data combinations performs extremely well in comparison to the GT approach, which is usually strongly disfavored. Using the combination of DESy5+DESI alone, we find that the GT approaches are disfavored at $|\log \mathcal{B}| \sim 5.8$ and $|\log \mathcal{B}| \sim 6.2$ for the Barrow and Tsallis limits on $\Delta$, respectively, wrt $\Lambda$CDM model. While the HDE approach is statistically equivalent to $\Lambda$CDM when comparing the Bayesian evidence. We also investigate the evolution of the dark energy equation of state and place limits on the same, consistent with quintessence-like behaviour in the HDE approaches.