A gas turbine engine includes a main compressor section having a downstream most location, and a turbine section, with both the main compressor section and the turbine section housing rotatable components. A first tap taps air compressed by the main compressor section at an upstream location upstream of the downstream most location. The first tap passes through a heat exchanger, and to a cooling compressor. Air downstream of the cooling compressor is selectively connected to reach at least one of the rotatable components. The cooling compressor is connected to rotate at a speed proportional to a rotational speed in one of the main compressor section and the turbine section. A valve system includes a check valve for selectively blocking flow downstream of the cooling compressor from reaching the at least one rotatable component. A dump valve selectively dumps air downstream of the cooling compressor. A method is also disclosed.