I've noticed that the player will run hot or cool depending on which AC power supply I've got it plugged into, and sometimes it will also run hot in the car. Is that because sometimes the car or the AC adapter is supplying more than 12-13 volts?

A switch mode PSU is still not 100% efficient, so there is a small amount of excess heat produced. The larger the differential between the input and output voltages, the more heat, hence the difference you notice with different mains adaptors. In addition, as I said, there are a couple of linear regulators which also produce some waste heat. Also, I suspect that the SMPSU chip runs more or less efficiently depending on the ripple and noise on the input supply, which would produce different amounts of heat with different, nominally identical output mains adaptors.

If the main regulators in the 3.3V and 5V supplies for the empeg were linear ones, it would get MUCH hotter, as it would be disippating around 16-20W of heat when running in the car. Also, don't forget, ultimately pretty much all the power put into electronics comes out as heat in the end, except for that radiated as light, RF, magnetism, or sound. Which all end up as heat in the long term anyway, just somewhere else. Such is entropy.

pca