We investigate the distribution and temperature-dependent optical properties of sharp, zero-phonon emission from defect-based single photon sources in multilayer hexagonal boron nitride (h-BN) flakes. We observe sharp emission lines from optically active defects distributed across an energy range that exceeds 500 meV. Spectrally-resolved photon-correlation measurements verify single photon emission, even when multiple emission lines are simultaneously excited within the same h-BN flake. We also present a detailed study of the temperature-dependent linewidth, spectral energy shift, and intensity for two different zero-phonon lines centered at 575 nm and 682 nm, which reveals a nearly identical temperature dependence despite a large difference in transition energy. Our temperature-dependent results are well described by a lattice vibration model that considers piezoelectric coupling to in-plane phonons. Finally, polarization spectroscopy measurements suggest that whereas the 575 nm emission line is directly excited by 532 nm excitation, the 682 nm line is excited indirectly.