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, Rényi, 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 $\mathrm{Pantheon}+\mathrm{and}$ 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 $\mathrm{\Lambda }\mathrm{CDM}$-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 $\mathrm{\Delta }\log \mathcal{B}\sim -6$, while the GT approach as a whole is significantly disfavored relative to the $\mathrm{\Lambda }\mathrm{CDM}$ model with at least $\mathrm{\Delta }\log \mathcal{B}\sim -8$ (${\mathcal{S}}_3$) to $\mathrm{\Delta }\log \mathcal{B}\sim -13$ (${\mathcal{S}}_4$), when using the DESy5 and DESI-DR2 datasets.

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• Cosmological parameters
• Cosmology
• Dark energy
• General relativity