Since the equilibrium composition of strange matter is crucial to the accurate calculation of its neutrino energy loss rates, we determine the composition by using the thermodynamic potentials (per unit volume) of the strange matter, and find that the electron abundance decreases with the strong interaction coupling constant or ba-ryon density. After deriving the formula of neutrino energy loss rates of the strange matter, we study the cooling of a homogeneous strange star with a thin crust in detail, and compare it with that of a neutron star. We conclude that the surface temperature of a young strange star is much less than that of a young neutron star of the same age, so one can distinguish between a strange star and a neutron star.