In this article, we consider the damped time-harmonic Galbrun’s equation which models solar and stellar oscillations. We introduce and analyze hybrid discontinuous Galerkin discretizations, which are stable and convergent for any polynomial degree greater or equal than one and are computationally more efficient than discontinuous Galerkin discretizations. Additionally, the methods are stable with respect to the drastic changes in the magnitude of the coefficients occurring in stars. The analysis is based on the concept of discrete approximation schemes and weak T-compatibility, which exploits the weakly T-coercive structure of the equation.