A cooling system for a gas turbine engine comprising a closed circuit containing a change-phase fluid, the closed circuit having at least one cooling exchanger configured to be exposed to a flow of cooling air for the change-phase fluid to release heat to the cooling air. A plurality of heat exchangers in are heat exchange relation with the change-phase fluid in the closed circuit, the plurality including at least a first heat exchanger configured to receive a first coolant from a first engine system for the change-phase fluid to absorb heat from the first coolant, and a second heat exchanger configured to receive a second coolant from a second engine system for the change-phase fluid to absorb heat from the second coolant. The system is configured so that the fluid at least partially vaporizes when absorbing heat from at least one of the first coolant and the second coolant and at least partially condenses when releasing heat to the cooling air. A method for cooling engine systems of a gas turbine engine is also provided.