A gas turbine engine includes a fan rotor, a compressor section, a combustor section, and a turbine section. The turbine section is positioned downstream of the combustor section. A fan drive turbine in the turbine section, and a shaft connects the fan drive turbine to the fan rotor. An inlet duct is connected to a cooling air source and connected to a cooling compressor downstream of the fan drive turbine. The cooling compressor is connected to an air source, and connected to a turning duct for passing compressed air in an upstream direction through the shaft. A method is also disclosed.