This study explores the interaction between non-extensive entropic FLRW cosmology and the power-law infationary model, with a focus on the overlap between the scalar spectral index "ns" and the tensor-to-scalar ratio "r". Based on a conjecture that non-extensive entropy alters the energy{momentum content of the cosmic fuid, the analysis examines how these overlaps shift with different model parameters and compares the fndings to those from Bekenstein{Hawking (BH) entropic cosmology. The study highlights the impact of Tsallis, Renyi, and Sharma-Mittal entropies, uncovering a signifcant correlation between "ns" and "r" that suggests a deeper connection in power-law infationary dynamics. The results demonstrate that non-extensive entropies not only enable viable infation with a graceful exit but also address limitations inherent in the standard BH entropic framework, emphasizing the importance of precise parameter estimation. It suggesting a unifed thermodynamic perspective in cosmological .studies
