A gas turbine engine (20; 100) comprising: a main compressor section having a downstream most location, and a cooling air tap at a location upstream of the downstream most location; a main turbine section (104, 106), wherein said main turbine section (104, 106) and said main compressor section are mounted within a housing, and have rotatable components; a first heat exchanger connected to said cooling air tap, and connected to receive fuel, the gas turbine engine configured to supply the fuel to a combustor in said gas turbine engine downstream of the first heat exchanger, such that said fuel may cool cooling air, from said cooling air tap, in said first heat exchanger; and a boost compressor connected to receive the cooling air downstream of said first heat exchanger and to deliver the cooling air to at least one of the rotatable components in at least one of said compressor and turbine sections (101, 104, 106), wherein: a high pressure tap taps power air downstream of said downstream most location and the gas turbine engine is configured to deliver the power air to a first boost turbine, said first boost turbine expanding the power air; the gas turbine engine is configured to supply said power air downstream of said first boost turbine to cool a low pressure turbine in said main turbine section (104, 106); a second boost turbine is configured to receive a portion of the power air downstream of said first boost turbine and to drive said boost compressor. a main compressor section having a downstream most location, and a cooling air tap at a location upstream of the downstream most location; a main turbine section (104, 106), wherein said main turbine section (104, 106) and said main compressor section are mounted within a housing, and have rotatable components; a first heat exchanger connected to said cooling air tap, and connected to receive fuel, the gas turbine engine configured to supply the fuel to a combustor in said gas turbine engine downstream of the first heat exchanger, such that said fuel may cool cooling air, from said cooling air tap, in said first heat exchanger; and a boost compressor connected to receive the cooling air downstream of said first heat exchanger and to deliver the cooling air to at least one of the rotatable components in at least one of said compressor and turbine sections (101, 104, 106), wherein: a high pressure tap taps power air downstream of said downstream most location and the gas turbine engine is configured to deliver the power air to a first boost turbine, said first boost turbine expanding the power air; the gas turbine engine is configured to supply said power air downstream of said first boost turbine to cool a low pressure turbine in said main turbine section (104, 106); a second boost turbine is configured to receive a portion of the power air downstream of said first boost turbine and to drive said boost compressor. a high pressure tap taps power air downstream of said downstream most location and the gas turbine engine is configured to deliver the power air to a first boost turbine, said first boost turbine expanding the power air; the gas turbine engine is configured to supply said power air downstream of said first boost turbine to cool a low pressure turbine in said main turbine section (104, 106); a second boost turbine is configured to receive a portion of the power air downstream of said first boost turbine and to drive said boost compressor.