Constraining the Generalized Uncertainty Principle (GUP) parameter is crucial for probing potential quantum gravity effects in regimes that extend beyond the Planck scale. In this study, we place bounds on the \(\beta \) parameter, associated with the widely studied quadratic GUP model, using existing experimental data from nuclear matter and results from chiral effective field theory ( \(\chi \) EFT) calculations. We also assess the compatibility of neutron star (NS) matter prediction based on those extracted from NS observations. The quadratic GUP model shares the same dispersion relation as a specific version of Double Special Relativity (DSR), establishing a connection between one of the rainbow gravity (RG) parameters and the quadratic GUP parameter. We then explore NS properties within the RG framework, defining \(X = E/E_p\) alongside \(\beta \) . Therefore, we calculate the predictions for slow-rotating NS using the RG effective metric and compare these results with existing observational data. From our analysis, we obtain an upper bound of \(\beta = 1.5 \times 10^{-7}\) based on nuclear matter and neutron star matter data. We also find a non-zero lower bound of \(\beta = -1.5 \times 10^{-7}\) . When using \(\beta = 1.5 \times 10^{-7}\) within the RG framework, the maximum mass prediction is lower than the constraints derived from the NICER data. In fact, rather than increasing, the parameter X further decreases the maximum mass prediction. However, when we set \(\beta = -1.5 \times 10^{-7}\) and \(X = 10^{-38.5}\) , the maximum neutron star mass remains consistent with NICER and other astrophysical constraints. Our results show that slowly rotating NS favor negative \(\beta \) within this framework.
