So there are two things to worry about with compressor life. The first, and most important is short-cycling. When a cooling circuit is running it builds up a lot of pressure on the output side of the compressor. When the cooling stops that pressure is still there until it has time to equalize out into the entire circuit. If you turn off a compressor and turn it back on before the pressure has equalized then the compressor has to go from a dead stop to overcoming a very high pressure output. This puts a lot of strain on the compressor and can lead to burning out the motor.
The second issue is simply the duty cycle and how long you're running the compressor in a given period of time. Fridges are well insulated and most are designed to be opened fairly rarely and then have time to recover back to their target temperature.
I haven't been able to track down hard and fast numbers but it's safe to say that anything less than a minute or two is quite likely to cause a lot of excess strain on your compressor on start-up. Three minutes is probably not going to kill your compressor but it's certainly on the short side. I personally use something in the 8-10 range just to be safe.
With all that said, I wouldn't recommend using a PID algorithm on a compressor driven cooling system. I don't think you're going to see much benefit in terms of temperature stability and the limitations of short-cycling mean that the PID can't even provide the very small adjustments that it expects to be able to make (you can't blip a compressor for 10 seconds to just provide a little bit of cooling, for example.)