One might ask, "what does the protect mode protect the amplifier from?"
There are usually 3 different things, maybe more, that the amp protects itself from by shutting down.
1. Excessive Heat
2. Excessive Voltage
3. Excessive Current

Being that you only have the one subwoofer hooked up, as far as I can tell, chances are that it is not because the amp cant handle the impedance, being that is either a 2,4,or 6ohm speaker.

Chances are that you do not have too much voltage goiing through the system, unless you connected a really long extension cord from your house to your car.

The weather is getting cooler now, at least by me. I doubt that the amp is overheated at start-up.

A couple things do come to mind though. A short circuit is likely, whether it is internal or in the wiring. The only way to find it is to check. Make sure that the 3 power wires are not touching at the terminals. You may have to go as far as to check the wiriing behind the head unit to make sure that none of them are touching either.

Oh yeah and hear is another quick test you can do. If you have a stereo system in your house, not a boombox or a shelf system, you can test to see if the problem is in the subwoofer. Most, if not all, home amplifiers have the same type of protection mode. Hook the subwoofer up, if it goes into protect mode, you found your problem. If you do try this method, make sure you have the volume low. This should really only be done if the speaker is 4+ohms, anything lower may make the indoor amplifier go into protect mode regardless. A Brute Force Method

If none of this does work than yes the one assumption was correct.... it is a shitty amp