Consider a nonlinear Klein-Gordon equation on a compact manifold M with smooth Cauchy data of size e → 0. Denote by T e the maximal time of existence of a smooth solution. One always has T e ≥ c e −k+1 if the nonlinearity vanishes at order k at 0. We prove that when M=Td, k=2, and the equation is quasilinear, T e ≥ ce −2 . When M=Sd−1, the equation is semilinear, and k = 2p, we show that T e ≥ ce −2p for almost every mass. The proof of these results relies on a systematic use of normal forms methods.