Rear hub cooling for high pressure compressor

A gas turbine engine (10) includes a turbine (24) and a high pressure compressor (18). The high pressure compressor (18) includes a last stage having a last stage compressor blade (34a) and a last stage vane (36a). The gas turbine engine (10) includes a first flow path through which bleed air flows to the turbine (24) and a second flow path through which air from the last stage of the high pressure compressor (18) flows. The bleed air in the first flow path exchanges heat with a portion of the air in the second flow path in a heat exchanger (50) to cool the air in the second flow path. The cooled air in the second flow path is returned to the high pressure compressor (18) to cool the high pressure compressor (18).