It doesnt cost much to bench test an amp. Your probably looking at about 25-30 bux. Have you checked the fuse on the power wire going to the amp? If its a glass fuse, the ends sometimes come loose and can mis connect, causing a loose intermittent connection. If its a wafer fuse change it out and put in a new one.
This happened to me although I did not have the same symptoms. it just quite powering up.
The sub thump you hear is probably more than likely turn on noise when your amp powers back up.
Check/replace the amp wire fuse and see if that takes care of the problem.
EDIT: If all else fails, you could have a bad soldering joint at the input. If you care to take apart the amp, you can easily check just by doing a visual inspection of the PCB near the input/output.