Actually it points to a subwoofer amp which wasn't powering down with the radio when it's supposed to, combined with a 5-year battery that was 8 years old, but thanks for the advice. The amp has been disconnected until I address it.
But I will remove the fuses and visually inspect them- I checked them with a low-current continuity checker in my multimeter which would not have caused a thermal situation like you describe.