The invention relates to a damping device for a gas turbine combustor with significantly reduced cooling air mass flow requirements. The damping device includes a wall with a first inner wall and a second outer wall, arranged in a distance to each other. The inner wall is subjected to high temperatures on a side with a hot gas flow. A plurality of cooling channels extend essentially parallel between the first inner wall and the second outer wall, and at least one damping volume bordered by cooling channels. Furthermore, the damping device includes a first passage for supplying a cooling medium from a cooling channel into the damping volume and a second passage for connecting the damping volume to the combustion chamber. An end plate, fixed to the inner wall, separates the damping volume from the combustion chamber.