Unveiling Phase Space Modifications: A Clash of Modified Gravity and the Generalized Uncertainty Principle

This study explores the link between Modified Gravity and modifications of phase space volume. Analyzing Fermi gas modifications {in the non-relativistic limit of the} Ricci-based gravities, we derive a generalized partition function in the grand-canonical ensemble, connecting Modified Gravity models with the Generalized Uncertainty Principle. Using this correspondence, we also establish bounds on the linear Generalized Uncertainty Principle: $-6\times {10}^{22}\lesssim \sigma \lesssim 3\times {10}^{22}\text{ s}\mathrm{/}\text{kg m}$, as well as lower bounds for Palatini $f(R)$ gravity $\beta >-7.51587\times {10}^7{\text{ m}}^2$ and Eddington-inspired Born-Infeld gravity $\epsilon >-1.88\times {10}^7{\text{ m}}^2$, ensuring microscopic stability. This connection also facilitates testing gravity proposals through tabletop experiments.