The system is probably low on charge. The low pressure cutout switch prevents the compressor from running that way because it could draw the low side into a vacuum. If there is a leak, air could be drawn in along with the humidity in it. That moisture combines with the oil in the system to form an acid that corrodes metal parts.
If the system still uses R-12, when it's at rest the pressure will be roughly equal to outside temperature as long as there is still a little liquid left. As the vapor leaks out, pressure goes down which allows more liquid to boil, vaporize, and expand, so the pressure goes right back up. If the system pressure is lower than outside temperature, there is no more liquid left to vaporize and expand. The same thing happens with R-134 but it is just a coincidence that system pressure and outside temperature match with R-12.
To prove if this is the problem, use a jumper wire to bypass the low pressure cutout switch.