Brillouin spectra have been collected in situ at temperatures up to ~1000 K for different crystallographic directions from two single crystal plates of LaAlO₃ perovskite. Elastic moduli derived from these, together with heat capacity, spontaneous strain and Raman data from the literature, have then been used to calibrate the coefficients in a classical Landau free energy expansion for the second order phase transition at T_c = 817 K. The static strain/order parameter coupling model provides a quantitative description of elastic softening between room temperature and ~700 K, but from ~700 K up to T_c additional elastic softening correlates with the development of a central peak in the Brillouin spectra. The presence of quasi-elastic scattering, which reaches maximum intensity ~5–15 K below T_c, implies a strong dynamical component to the phase transition. Relaxation times estimated from the width of the central peak are of the order of ~10–100 ps and appear to be more or less constant between ~700 and 800 K, which is consistent with an intrinsic origin associated with phonon density fluctuations. Central peak width variations and an irregular pattern of acoustic velocity variations in a 20 K temperature interval below T_c are interpreted in terms of flipping of clusters of tilted octahedra between different 111, 011 and 001 tilt axes. The additional softening beyond that expected from the classical strain/order parameter coupling model can be understood in terms of coupling of acoustic modes with the central peak mode(s).