In one aspect the present subject matter is directed to a system for cooling a turbine engine. The system includes a cooling medium source, a stator vane having an internal flow passage that is in fluid communication with the cooling medium source and a turbine shroud assembly having an internal flow passage that is in fluid communication with the internal flow passage of the stator vane. The system allows for reduced peak thermal gradients between a cooling medium provided by the cooling medium source and various turbine hardware components such as the turbine shroud assembly.