A compressor rotor compresses air and delivers the compressed air to a downstream inlet and across a turbine rotor. A shaft rotates with the turbine rotor and the compressor rotor. The shaft is hollow with an inner bore and an outer periphery. At least one journal bearing supports a portion of the shaft, and has an inner bore spaced from the outer periphery of the shaft. A cooling air path is provided between the shaft outer periphery and the bearing inner periphery, and along a length of the bearing to at least one connection hole. The connection hole provides cooling air from an outer periphery of the shaft to the inner bore of the shaft. In a separate feature, a shaft for use in an air supply machine includes a hollow interior, with such a connection hole.