Your interpretation of the results. I agree with you that if you find no imaginary frequencies in the cubic phase it means it is at a local minimum of the potential energy landscape, and that if you do find imaginary frequencies for the tetragonal phase, then that one is at a saddle point. I also agree that a phase exhibiting imaginary frequencies may be stabilized by entropic contributions (see more below). Just a small caveat at this point: entropic contributions typically favor higher symmetry phases, so I am a little surprised that you find a stable cubic phase but an unstable tetragonal phase. This suggests that the cubic phase is the low temperature phase, while the tetragonal phase is the high temperature phase. In many compounds (e.g. perovskites) this is usually the other way around. Either way, the comments below apply to any unstable phase.
Finite temperature stabilization. While you are correct that a phase exhibiting imaginary phonons may be stabilized by entropic finite temperature contributions, your approach to evaluate these looks incorrect. If I understand what you are doing, then you are evaluating the Helmholtz free energy as a function of temperature using the harmonic approximation. While this is fine for a phase without imaginary phonons, this is incorrect for a phase with imaginary phonons. How would you include the contribution of the imaginary modes to the free energy? Instead, what you need to do is to include anharmonic terms in the expansion of the potential energy landscape. This is the only way in which you may be able to accurately evaluate the Helmholtz free energy of a phase that exhibits imaginary phonons at the harmonic level.
Anharmonic calculations. There are a variety of strategies to include anharmonic terms in the calculations to evaluate the Helmholtz free energy. Methods range from molecular dynamics to the self-consistent harmonic approximation, and which one is more appropriate will depend on the details of your system. However, their computational expense is significantly larger than that of harmonic calculations.