A cooled hybrid structure is provided for deployment within a gas turbine engine. In one embodiment, the cooled hybrid structure includes a woven oxide fiber sheet and an insulative oxide coating. The woven oxide fiber sheet includes an outer cold wall and an inner hot wall, which is integrally woven with the outer cold wall and which cooperates therewith to define a plurality of elongated cooling channels extending within the woven oxide fiber sheet. A plurality of impingement apertures is formed through the outer cold wall and conducts airflow into the plurality of elongated cooling channels and against the inner hot wall to convectively cool the woven oxide fiber sheet. A plurality of effusion channels is formed through the inner hot wall and through the insulative oxide coating and conducts airflow through the insulative oxide coating to provide convective cooling thereof.