A turbomachine includes a compressor configured to compress air received at an intake portion to form a compressed airflow that exits into an outlet portion. The compressor has a plurality of rotor blades and a plurality of stator vanes, and a compressor casing forming an outer shell of the compressor. A combustor is operably connected with the compressor, and the combustor receives the compressed airflow. A turbine is operably connected with the combustor, and the turbine receives combustion gas flow from the combustor. The turbine has a turbine casing. A cooling system is operatively connected to the compressor casing. The cooling system includes a plurality of heat pipes located in at least a portion of the plurality of stator vanes. The heat pipes are configured to be in thermal communication with the compressor casing. The heat absorbed by the plurality of heat pipes is transferred to the compressor casing.