The temperature dependence of the elastic properties of antiferroelectric PbHfO3 was investigated by Brillouin scattering. The two structural phase transitions of antiferroelectric-antiferroelectric-paraelectric phases were clearly identified by discontinuous changes in the acoustic mode frequencies and the hypersonic damping. The substantial softening of the mode frequency along with the remarkable increase in the acoustic damping observed in the paraelectric phase indicated the formation of precursor noncentrosymmetric (polar) clusters and their coupling to the acoustic waves. This was corroborated by the observation of quasi-elastic central peaks, the intensity of which grew upon cooling toward the Curie point. The obtained relaxation time exhibited a slowing-down behavior, suggesting that the dynamics of precursor clusters becomes more sluggish on approaching the phase transition temperature.