In order to improve the cooling of an air-cooled gas turbine (10) in the partial load operating mode it is proposed to provide a connecting line (7) between two cooling air lines (5, 6) with different pressure levels, which connecting line (7) leads from the second cooling air line (6) at a relative high pressure level to the first cooling air line (5) at a relative low pressure level. In this context, a cooling device (9) for cooling an auxiliary cooling air stream, flowing from the second cooling air line (6) into the first cooling air line (5), and an adjustment element (11) are arranged in the connecting line (7). In addition to a gas turbine (10), a method for operating such a gas turbine (10) is the subject matter of the disclosure.