We study universality in the kinetics of spinodal phase separation in unstable thin liquid films, via simulations of the thin film equation. It is shown that, in addition to morphology and free energy, the number density of local maxima in the film profile can also be used to identify the early, late, and intermediate stages of spinodal phase separation. A universal curve between the number density of local maxima and rescaled time describes the kinetics of the early stage in d=2 and 3. The Lifshitz-Slyozov exponent of -1/3 describes the kinetics of the late stage in d=2 even in the absence of coexisting equilibrium phases.