That's a good point. If the problem turns out to be a component
inside the amp instead of just a blown fuse, it could have easily
happened because of the "inrush curent" through cold circuits when you
turned the amp on. (The same reason light bulbs usually blow when you
first turn them on.)

Amps are usually protected from voltage spikes and drops while you're
starting your car, since the amp is turned on by the remote turn-on
lead on the head unit, and there's usually a relay in your car's
wiring that keeps things like the air conditioning and head unit
turned off while the engine is being started.

But with the light bulb, it's because the resistance of the tungsten is at
its lowest point when cold. Things are different with an amp. Charge up
the power supply capacitors and that's it. What's gonna go wrong there?
Burn out a trace?