Maybe your "checked for leaks" wasn't as thorough as you thought? Regular cycling like that is an almost sure sign of a steady leak somewhere, but maybe not so easy to find.
Is there a place near the pump where you could close of the air line to eliminate the rest of the system (air suspension)? If you can do that, it isolates the pump and its immediate plumbing from the larger system. If the cycling still occurs, suspicion falls on the pump and its fittings and you don't have to worry about the rest. But if the cycling stops, you know the pump is OK and the suspension leaks somewhere.
Former owner of 2004 American Tradition and several other RVs
Home is in the Ocala Nat'l Forest near Ocala, FL
Summers in Black Mountain, NC