A method of determining the temperature of the nonradiative reservoir in a microcavity exciton-polariton system is developed. A general relation for the homogeneous polariton linewidth is theoretically derived and experimentally used in the method. In experiments with a GaAs microcavity under nonresonant pulsed excitation, the reservoir temperature dynamics is extracted from the polariton linewidth. Within the first nanosecond the reservoir temperature greatly exceeds the lattice temperature and determines the dynamics of the major processes in the system. It is shown that, for nonresonant pulsed excitation of GaAs microcavities, the polariton Bose-Einstein condensation is typically governed by polariton-phonon scattering, while interparticle scattering leads to condensate depopulation.