A gas turbine engine comprises a compressor section and a turbine section, the compressor section having a last compressor stage. High pressure cooling air is tapped from a location downstream of the last compressor stage and passed through a heat exchanger. Lower pressure air passes across the heat exchanger to cool the high pressure cooling air. A housing surrounds the compressor section and the turbine section and there being a space radially outwardly of the housing, and a mixing chamber received in the space radially outwardly of the housing, the mixing chamber receiving the high pressure cooling air downstream of the heat exchanger, and further receiving air at a temperature higher than a temperature of the high pressure cooling air downstream of the heat exchanger. Mixed air from the mixing chamber is returned into the housing and utilized to cool at least the turbine section.