The invention discloses the multistage cooling of an aircraft electronic system (2) with at least one electronic component which delivers heat. The multistage feature results from the use of a plurality of circuits for transferring the waste heat. An internal coolant (14) circulating in a closed circuit which is thermally coupled to the electronic component carries heat from the at least one electronic component to a heat exchanger (6) which delivers the heat to an external coolant which flows and/or circulates from a source outside of the aircraft electronic system (2) through the heat exchanger (6) to a sink outside of the aircraft electronic system (2). The internal coolant flows from the heat exchanger (6) in the direction of the at least one electronic component and/or circulates in a closed circuit.