Bob wrote:

I have a 20-year old Frigidaire through the wall air conditioner.

When first turned on it does a great cooling job. A digital
thermometer right at the outlet is showing 40°F.

After three to five minutes, the compressor turns off and the cooling
stops.

After maybe 5 minutes the compressor comes back on and cools for
another 3 to 5 minutes.

I took the unit outdoors and cleaned the fins with a garden hose and a
fin-cleaning aerosol spray. It is difficult to view the fins
visually, but I'm fairly sure they are clean. The cleaning did not
seem to change the cycling time at all.

I'm in a small town and it is a heat wave. It is difficult to get a
repair person in the best of times---and I'd rather save the money and
repair it myself if it is something I can handle.

How do I troubleshoot from this point?

I'm with Ed. You should be able to get a new unit that will pay for
itself fairly quickly in energy savings compared to a 20 year old
unit.