A gas turbine engine has a compressor section with a downstream most end and a cooling air tap at a location spaced upstream from the downstream most end. The cooling air tap is passed through at least one boost compressor and at least one heat exchanger, and then passed to a turbine section to cool the turbine section. The boost compressor is driven by a driveshaft which is driven by the turbine section. A boost turbine selectively drives the boost compressor.