First of all the item you refer to isn't a pump it's a compressor and if the fridge is not cooling it is almost certain that the compressor has failed.
You can find the compressor at the rear of the fridge and if working properly the casing will be hot to the touch. Only check this after you have disconnected the power supply. Compressors are usually sealed for life and are not therefore repairable.It will be difficult to check if the thermostat is the cause of the problem therefore you should ideally call in a qualified refrigeration service engineer.
Roger
January 2008