What is a gas turbine cooled by?

It depends on the type of engine, but its generally one of a few things. A turbofan like on a commercial plane have the bypass, or large amounts of air flowing close to the out side of the engine. A higher temperature engine may have a heat exchanger on the turbine blades that actuall takes cooler air and passes it over the vanes so they don't melt. Almost every turbine engine has some sort of oil cooler to help keep the moving parts from getting too hot. With many turbines the idea is to more or less protect the engine from heat than to cool it down. More heat with a turbine means more power.