A flange cooling structure of a gas turbine engine in which a compressor, a combustor, and a turbine are arranged along a rotating shaft, the flange cooling structure comprising: a first member including a first main body (11a, 12a) and a first flange (11b, 12b), the first main body (11a, 12a) defining a first space, the first flange (11b, 12b) projecting from the first main body (11a, 12a), heat of the combustor being transferred to the first member; a second member including a second main body and a second flange, the second main body defining a second space between the first main body and the second main body, the second flange projecting from the second main body, the second member being lower in temperature than the first member during operation of the gas turbine engine; and fasteners that fasten the first flange (11b, 12b) and the second flange to each other to form a flange laminated body that divides the second space, wherein: the second space is lower in temperature than the first space during the operation of the gas turbine engine; the second space includes a first region (S2a) located at one side of the flange laminated body and a second region (S2b) located at the other side of the flange laminated body; the first region (S2a) is higher in pressure than the second region (S2b) during the operation of the gas turbine engine; one of the first flange (11b, 12b) and the second flange is a high pressure flange that faces the first region (S2a); and the other of the first flange (11b, 12b) and the second flange is a low pressure flange (11b, 12b) that faces the second region (S2b), wherein the first member includes a housing of the combustor and a stationary blade unit of the turbine; wherein the second member includes a combustor case arranged outside the combustor; wherein the first flange (11b, 12b) includes a combustor flange of the combustor and a stationary blade unit flange of the stationary blade unit; and wherein the second flange includes a case flange of the combustor case, characterized in that a contact surface of the high pressure flange or a contact surface of the low pressure flange (11b, 12b) includes a cooling groove that communicates with the first region (S2a) and the second region (S2b). a first member including a first main body (11a, 12a) and a first flange (11b, 12b), the first main body (11a, 12a) defining a first space, the first flange (11b, 12b) projecting from the first main body (11a, 12a), heat of the combustor being transferred to the first member; a second member including a second main body and a second flange, the second main body defining a second space between the first main body and the second main body, the second flange projecting from the second main body, the second member being lower in temperature than the first member during operation of the gas turbine engine; and fasteners that fasten the first flange (11b, 12b) and the second flange to each other to form a flange laminated body that divides the second space, wherein: the second space is lower in temperature than the first space during the operation of the gas turbine engine; the second space includes a first region (S2a) located at one side of the flange laminated body and a second region (S2b) located at the other side of the flange laminated body; the first region (S2a) is higher in pressure than the second region (S2b) during the operation of the gas turbine engine; one of the first flange (11b, 12b) and the second flange is a high pressure flange that faces the first region (S2a); and the other of the first flange (11b, 12b) and the second flange is a low pressure flange (11b, 12b) that faces the second region (S2b), wherein the first member includes a housing of the combustor and a stationary blade unit of the turbine; wherein the second member includes a combustor case arranged outside the combustor; wherein the first flange (11b, 12b) includes a combustor flange of the combustor and a stationary blade unit flange of the stationary blade unit; and wherein the second flange includes a case flange of the combustor case, characterized in that a contact surface of the high pressure flange or a contact surface of the low pressure flange (11b, 12b) includes a cooling groove that communicates with the first region (S2a) and the second region (S2b). the second space is lower in temperature than the first space during the operation of the gas turbine engine; the second space includes a first region (S2a) located at one side of the flange laminated body and a second region (S2b) located at the other side of the flange laminated body; the first region (S2a) is higher in pressure than the second region (S2b) during the operation of the gas turbine engine; one of the first flange (11b, 12b) and the second flange is a high pressure flange that faces the first region (S2a); and the other of the first flange (11b, 12b) and the second flange is a low pressure flange (11b, 12b) that faces the second region (S2b), wherein the first member includes a housing of the combustor and a stationary blade unit of the turbine; wherein the second member includes a combustor case arranged outside the combustor; wherein the first flange (11b, 12b) includes a combustor flange of the combustor and a stationary blade unit flange of the stationary blade unit; and wherein the second flange includes a case flange of the combustor case, characterized in that a contact surface of the high pressure flange or a contact surface of the low pressure flange (11b, 12b) includes a cooling groove that communicates with the first region (S2a) and the second region (S2b).