A gas turbine engine includes a heat exchanger, a diffuser case, a passageway and a nozzle assembly. The heat exchanger exchanges heat with a bleed airflow to provide a conditioned airflow. The diffuser case includes a plenum that receives the conditioned airflow. The passageway is fluidly connected between the heat exchanger and the diffuser case, and the conditioned airflow is communicated through the passageway and into the plenum. The nozzle assembly is in fluid communication with the plenum of the diffuser case to receive the conditioned airflow from the plenum.