AC cycling but only in extreme heat

Greetings. AC cycling has been discussed before (that's how I found this forum) but my problem may be a little more specific and I'd like some direction.

I recently inherited a 2002 Malibu LS. I didn't notice any problems with the AC until it's been very warm here. Then the AC cycles on and off for the first half hour of driving. The car sits on a parking lot all day and today the heat index was 109. So the car is hot. The AC blows cold air when it's on but it shuts itself off and automatically comes back on while driving. By "off", I mean that the light on the AC switch goes off and warm air blows from the vents. The blower runs normally throughout the cycling. By the time I get home, it's usually settled down and stays on. There doesn't seem to be any correlation to engine or vehicle speed. Sometimes it will stay on while waiting at a light. Other times it shuts off and on at 60 mph. It appears that once enough (albeit hot) air blows over the engine, something cools down enough and the AC starts working as it's supposed to. Today, I noticed that if the AC light is on, as soon as I press the button to recycle interior air, the AC shuts off for a few seconds then comes back on with the outside air light on. Try to switch it to interior recycled air and the same thing happened again. It did this several times before I gave up and let it do its own thing.

Does this sound familiar to anyone? Is there something in the system that would shut down in extreme heat but then start working again once it's cooled down?

TIA