Our 2003 Ford Escort ZX2 sounds its horn continually when the temperature drops into the low 40’s, and can only be stopped by pulling the fuse. Pushing the horn button sometimes stops it for a few minutes, but it starts up again. It is a continuos sound, not the alarm beep.
I think you have found the problem area when you push the horn button/pad and the horn stops sounding. When the car gets cold the horn pad is contracting and bringing the contacts into continuity. If you can get to the wires connected to the pad, disconnect them on a night when the horn would most likely sound and see if the problem is solved. It sounds like a replacement of the horn pad is needed.