Washing Machine Overheating

It sounds like it could be the thermostat, but not necessarily. Thermostats are easy enough to replace. There is the possibility of it being a programme timer fault though if the thermostat closes but the timer doesn't respond, or even a wire off the thermostat, or electrical connection somewhere in the thermostat circuit. You'd need to be prepared to gamble that it is the thermostat.

UPDATE:

These days, most washing machines should not overheat. For many years, if a fault occurred, they could boil the water dry (which was ridiculous) but a modern washing machine should abort the program and give an error code rather than continue heating the water after a heating fault. Also, most washing machines now control the water temperature not with a bi-metal-type thermostat, but with a thermistor. A thermistor is a device filled with crystals that alter their electrical resistance in a predictable way according to their temperature. Thermisters tend to be pretty reliable.

Related:

Washing machine stops with an error code | Program selector knob (dial) clicks round continuously

Another symptom with a faulty thermistor can be that the machine can be put into a constant cycle of filling with cold water and pumping the water away if it thinks the washing machine is overheating. The logic of this is that if the water is far too hot, then by pumping and topping up with cold, the temperature can be brough back to safe levels. I've seen this happen on Indesit washing machines. However, do not confuse this with the much more common siphoning fault which also causes the washer to fill and drain the water in a cycle - Washing machine constantly fills with water and goes down the drain