Structures, such as compressor casings, for reducing a thermal gradient are provided. For example, a compressor case is split such that it includes first and second case segments. The first case segment extends over a first portion of the compressor case circumference and comprises a first inner structure, a first outer structure, and a first porous structure integrally formed as a monolithic component. The first porous structure is defined between the first inner structure and the first outer structure. The second case segment extends over a second portion of the compressor case circumference and comprises a second inner structure, a second outer structure, and a second porous structure integrally formed as a monolithic component. The second porous structure is defined between the second inner structure and the second outer structure. Methods of cooling structures such as compressor casings also are provided.