The heating elements on this heater just kept on heating and heating independent of the temperature. I first checked the thermocouple that measures temperature - but that seemed OK, so the next thing I suspected was the solid-state relay that turns on/off the heating. Turns out this initial guess was right, and I was lucky we had a replacement on the shelf, so now the heater works again. The picture above shows the broken SSR on the right.
New SSR in place.