Unlike in thermodynamic equilibrium where coexisting phases always have the same temperature, here we show that systems comprising "active" self-propelled particles can self-organize into two coexisting phases at different kinetic temperatures, which are separated from each other by a sharp and persistent temperature gradient. Contrasting previous studies which have focused on overdamped descriptions of active particles, we show that a "hot-cold-coexistence" occurs if and only if accounting for inertia, which is significant in a broad range of systems such as activated dusty plasmas, microflyers, whirling fruits or beetles at interfaces. Our results exemplify a route to use active particles to create a self-sustained temperature gradient across coexisting phases, a phenomenon, which is fundamentally beyond equilibrium physics. arXiv:1902.06116v2 [cond-mat.soft]