We study zero and finite temperature static Bose-Einstein condensate (BEC) stars in the combined Rastall-Rainbow (RR) theory of gravity by considering different BEC equation of states (EoSs). We obtain the global properties of BEC stars by solving the modified Tolman-Oppenheimer-Volkoff equations of RR gravity with values of Rastall parameter \(\kappa \) and Rainbow function \(\Sigma \) chosen accordingly. We observe that the parameter \(\kappa \) has negligible effect on the maximum mass of the stars considered, whereas \(\Sigma \) alters it significantly, and increasing the value of \(\kappa \) beyond a certain limit results in unstable solutions for any value of \(\Sigma \) . We report that the inclusion of temperature in our analysis expands the parameter space by including more values of \(\kappa \) . However, temperature has negligible effect on the maximum mass of the stellar profiles in all the three theories. We have also studied the compactness and stability of the obtained stellar equilibria. We report that BEC stars satisfy various energy conditions within the range of \(\kappa \) and \(\Sigma \) taken in our paper. Further, we find that the maximum masses and radii of the stars within RR theory can have good agreement with the observational data on pulsars for all the EoSs considered and in particular, the Colpi-Wasserman-Shapiro EoS, which was ruled out in General Relativity (GR). We also find that, in contrast to the results of GR, BEC stars consistent with observations can be realised in the RR theory with smaller bosonic self-interaction strength.
