The convergence of the kinetic Langevin simulated annealing is proven under mild assumptions on the potential $U$ for slow logarithmic cooling schedules. Moreover, non-convergence for fast logarithmic and non-logarithmic cooling schedules is established. The results are based on an adaptation to non-elliptic non-reversible kinetic settings of a localization/local convergence strategy developed by Fournier and Tardif in the overdamped elliptic case, and on precise quantitative high order Sobolev hypocoercive estimates.