A rotary machine cooling system is provided that includes a housing having an inlet and outlet. A rotor supported in the housing for rotation about an axis with the rotor including an outer surface. A stator is disposed within the housing about the rotor having an inner surface adjacent to the outer surface of the rotor. A gap is formed between the inner and outer surfaces sufficient to reduce friction and permit convection cooling. The gap is in fluid communication with the inlet and outlet, and a pumping device pumps a cooling gas from the inlet into the gap and through said outlet. The pumping device includes an impeller affixed to an end portion of the rotor to create an axial airflow into the gap. Preferably, the present invention air cooling system is used in conjunction with a liquid cooling system arranged exteriorly to the stator. The liquid cooling system reduces the amount of air needed to flow between the gap to remove the desired amount of heat from the machine.