Constraints on Generalized Gravity-Thermodynamic Cosmology from DESI DR2

We explore the cosmological implications of generalized entropic models within the framework of Gravity-Thermodynamics (GT) approaches. These models, characterized by three or four additional free parameters, are designed to capture deviations from the standard Bekenstein-Hawking entropy and can reproduce well-known entropic formulations, including Tsallis, R\'enyi, Sharma-Mittal, Barrow, Kaniadakis, and Loop Quantum Gravity entropies in various analytical limits. We implement the corresponding cosmological models using a fully numerical GT approach to constrain the model parameters and to study the evolution of the dark energy equation of state as a function of the scale factor. Our Bayesian analysis, which incorporates the Pantheon+ and DESy5 supernovae data alongside the recently released DESI-DR2/DR1 Baryon Acoustic Oscillation (BAO) measurements, shows that the data favor the standard Bekenstein-Hawking entropy, leading to a $\Lambda$CDM-like late-time behavior. In this context, the three-parameter ($\mathcal{S}_3$) entropic model appears to be sufficient to capture the observed dark energy phenomenology. Furthermore, a direct comparison of the Bayesian evidence indicates that the three-parameter model is preferred over the four-parameter ($\mathcal{S}_4$) variant by a factor of $\Delta\log\mathcal{B} \sim -6$, while the GT approach as a whole is significantly disfavored relative to the $\Lambda$CDM model with at least $\Delta\log\mathcal{B} \sim -8$ ($\mathcal{S}_3$) to $\Delta\log\mathcal{B} \sim -13$ ($\mathcal{S}_4$), when using the DESy5 and DESI-DR2 datasets.