The understanding of the elastic constants could help you to figure out how is changing the material previous to the phase transition. Also, this could be related wit a softening of a phonon mode.
Usually the Youn's modulus increases with pressure, but sometimes it decreases with increasing pressure, please see for example in: O. Gomis, F.J. Manjón, P. Rodríguez-Hernández, A. Muñoz, Elastic and thermodynamic properties of α-Bi2O3 at high pressures: Study of mechanical and dynamical stability, Journal of Physics and Chemistry of Solids