A gas turbine combustor (26) disposed in a combustion air plenum (65) and a transition piece (28) for the combustor disposed in a separate cooling air plenum (58, 67). The combustion air plenum may receive combustion air (50) from a high-pressure compressor stage (22A). The cooling air plenum may receive cooling air (52) from an intermediate-pressure compressor stage (22B) at lower temperature and pressure than the combustion air. This cools the transition piece using less air than prior systems, thus making the gas turbine engine (20) more efficient and less expensive, because less expensive materials are needed and/or higher combustion temperatures are allowed. The cooling air may exit the cooling air plenum through holes (62) in a downstream portion (61) of the transition piece. An outer wall (72) on the transition piece may provide forced convection along the transition piece.