Exploring Cosmic Evolution in R\'enyi Entropic Cosmology with Constraints from DESI DR2 BAO and GW Data

We explore a cosmological model based on R\'enyi entropic corrections to the Friedmann equations and constrain it using DESI, P-BAO, CC, and gravitational-wave observations. Unlike earlier works, we obtain a direct and stringent constraint on the R\'enyi parameter $\lambda$ from late-time cosmic acceleration, with the resulting value even satisfying recent Big Bang Nucleosynthesis and baryogenesis bounds. The model predicts values of $H_0$ and $\Omega_{m0}$ that are consistent with current observational data. It provides a successful description of late-time acceleration with a quintessence-like behavior, smoothly approaching the $\Lambda$CDM limit without crossing the phantom divide. Furthermore, the statical comparisons along with the evaluation of energy conditions and stability analyses reinforce its viability as a robust alternative to the standard cosmological model.