I have an Amana C/A that runs normally during the day but late in the evening when the outside temp is the same as the inside temp the compressor turns off and then back on again in short cycles. We had a repairman out and he said it was the timer on the unit and that we didn't need it as we had a similar one in our thermostat. But I think he was mistaken as the unit continues to shut off when the temp outside cools and he removed the delay timer when the compressor shuts off to keep it from coming back on to quickly. We did have another repairman out earlier in the summer and he had to add refigerant (about 3 lbs) as the compressor would try to start and then shut off. The unit is about 9 years old.

Have a new time delay installed to solve the problem. If you have a new digital thermostat some of them can be adjusted to cycle the A/C unit at different intervals.

Have a new time delay installed to solve the problem. If you have a new digital thermostat some of them can be adjusted to cycle the A/C unit at different intervals.
I don't thinkl the time delay is the problem because when the compressor shut off before the time delay was removed the compressor would quit and then after a couple of minutes the compressor would start and then quit again. Now when the compressor shuts down it restarts almost immediately and then quits again. I do not have a new digital thermostat I have an old one from honeywell (chronotherm III) where the AC cycle interval is set at the factory. And why would this happen only when the out side temp has cooled down. Any other thoughts?

(chronotherm III) where the AC cycle interval is set at the factory

Yes

And why would this happen only when the out side temp has cooled down. Any other thoughts?

First of all your thermostat could have become defective.
Second when you start adding 3 pounds of refrigerant to a system it could be overcharged?

You really should have a time delay. This way if the thermostat calls for cooling right after the outside unit was on there will be a 5 minute time delay.

You may have a out of range low pressure cutout if the unit is so equipped.
You could try and track down what is turning off and on the 24volt low voltage control circuit but you would have to know what you were doing.

Depending where the thermostat is located can also be a problem especially if it is getting a warm draft across it. You might check to see if the installer caulked the area where the wires come through the wall to the thermostat. I have sen many times that the warm air in the wall will flow out and into the thermostat area and cause the unit to go off and one too much.

These are just some ideas but the time delay you need.