Interesting. I know, for example, with HID lighting, that once the lamp is turned off, you need to wait for it to cool down before it will re-start. I wasn't aware of anything in an amp that would require a cooldown, but then I know very little about electronics.
Not *all* HID lighting works like that. Automotive HID (Xenon) headlamps do not require any kind of cooldown/restart period. None that I've ever encountered, anyway.
But back to topic...
Neither my old Kenwood sub, current SVS, nor my Emotiva amp has any sort of 'power-off-timeout' feature. None of them produce any thumps when turning on or off either. I've certainly heard of amps going into a thermal or circuitry protection modes, but never anything like this.
Where's Alan when you need him?!