The linear stability of a rotating flow in an elliptically deformed cylindrical shell with an imposed radial temperature contrast is studied using local and global approaches. We demonstrate that (i) a stabilizing temperature profile can either increase or decrease the growth rate of the elliptical instability depending on the selected mode and on the strength of the radial buoyancy force; (ii) when the temperature profile is destabilizing, the elliptical instability coexists with two-dimensional convective instabilities at relatively small values of the Rayleigh number, the fastest growing mode depending on the relative values of the Rayleigh number and of the eccentricity; (iii) the elliptical instability totally disappears for larger values of the Rayleigh number. We argue that thermal effects have to be taken into account when looking for the occurrence and influence of inertial instabilities in geophysical and astrophysical systems, especially in planetary cores.