Check the voltage at the amplifier primary terminals during operation.
Is it a less expensive amplifier without a self-protection circuit? Could be a bad amplifier.
Check the security of all primary connections at the power source and amplifier.
Bench check the amplifier out of the boat and off the boat system.
Unplug the RCAs and see if it still turns on. Maybe its dependent on the 'above ground' RCA shields because of a poor ground.
The repetitive 'thump' is usually kind of a power oscillation where the internal voltage sags and the amplifier turns off followed by a voltage rise where the amplifier turns back on and it rapidly continues to repeat the cycle.