The pilot reported that while in the local traffic pattern, he was distracted by radio transmissions from the tower air traffic controller regarding maintaining his airspeed and multiple landing runway reassignments. His pre-landing check was interrupted, and he did not lower the landing gear, and landed gear up on the runway. The airplane sustained substantial damage to the undercarriage. The pilot reported that there were no preaccident mechanical malfunctions or failures with the airplane that would have precluded normal operation of the airplane. The pilot reported that he was wearing noise-cancelling headsets and did not hear the warning horn. During the airplane recovery, the pilot reported that the throttle was in the retarded position and the warning horn was blaring. The landing gear handle was placed in the “down” position and extended normally.