A cooling system for a gas turbine is provided that may include a drive shaft, a shield of a combustor of the gas turbine engine, and a conduit. The drive shaft may be configured to mechanically couple a compressor of the gas turbine engine to a turbine of the gas turbine engine. The shield may be positioned between the compressor and the turbine. The combustor may be configured to drive the turbine. The conduit may be configured to supply a cooling fluid to a gap between an outer surface of the drive shaft and the shield of the combustor.