A cooling device for a stationary ring surrounding a gas-turbine hot-gas flow path, said ring being made up of a plurality of ring segments that are fastened by upstream and downstream fastener systems onto a plurality of spacer segments forming a support spacer surrounding said ring so as to co-operate therewith to define at least one annular impact cavity into which at least one air supply orifice opens out, each ring segment having walls that are pierced with a plurality of air exhaust holes opening out both into the impact cavity, and into the hot-gas flow path, the device further comprising means for channeling the air coming from leakage through the seals of the fastener systems, and for directing it towards at least one of the axial ends of the ring segments so as to cool said end(s).