A cover (54, 54A-B) enclosing with clearance (65) an acoustic damping resonator (24) on a working gas path liner (22) of a gas turbine component (28). The cover includes a coolant inlet chamber (56, 56B) with a top wall (58, 58B) that is closer to the liner than a top wall (32) of the resonator, and extends upstream from the resonator relative to the working gas flow (48). Compressed air (26) surrounds the cover at a higher pressure than the working gas and flows (44) into and through the coolant inlet chamber, then through holes (34) in the resonator, then exits through holes (38) the liner into the working gas. The coolant inlet chamber directs the flow of compressed air over a weld (50) of the upstream wall (40) of the resonator to cool it. The cover may be formed as a box (57) or a sleeve (69).