This is most likely to be due to a faulty control board, as thats responsible for processing the input from the thermostat and then calling the compressor to start when it needs coolant.
If you can advise me of the model number I can locate the PCB for you, if you do chose to replace the PCB please ensure you order a pre-programmed one.
I had a look online and I cant see any of the usual firms offering a replacement control board \ PCB, however they normally cost around £100.
I would be inclined to replace the fridge rather than fix it, since you are now on a slippery slope of paying good money after bad as new faults develop.
Should you want me to recommend an engineer in your area to come out and repair it please provide your postcode.
Please let me know if I can be of any further assistance, appreciate if you can rate the service received to date.