That's the more unusual of the two scenarios.
What's the value of that breaker? It should be dedicated to the A/C alone.
Sometimes, breakers just get old and "weak", i.e. they trip easier than they should.
The easiest and cheapest thing to try is to buy another breaker. Take the old one with you to Home Depot or similar place, and just get another. They're relatively cheap, usually at about $6.00.
After that, we need to start investigating the run capacitor inside the A/C unit itself.
1999 Flagstaff 21FB, 2000 F-350 Lariat LB/CC 7.3L SRW 3:73