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We study the linear and nonlinear acoustic response of SrTiO3 across its ferroelastic transition at T-a = 105 K by time domain Brillouin scattering. Above T-a we observe that for a strain amplitude of similar to 0.18% the sound velocity for compressive strain exceeds the tensile strain velocity by 3%. Below T-a we find a giant slowing down of the sound velocity by 12% and attribute this to the coupling of GHz phonons to ferroelastic twin domain walls. We propose a new mechanism for this coupling on the ultrafast time scale, providing an important new test ground for theories used to simulate atomic motion in domain forming crystals.