Originally Posted by Maladjusted
Could be the pump itself is failing. The pump has a pressure switch and a backflow preventer.. to maintain the pressure until a faucet is opened. If the backflow valve is failing, it will allow the pressure to bleed off and the pump will kick on. I've had this happen on a boat water system.
A leak within the pump can cause the pressure switch to think it should turn on, and it does, but almost immediately the pressure builds and the pressure switch turns off the pump.
If that is the case you will not see any leaking water.
New diaphragms, repair kits and pressure switches are available for most RV water pumps at much less cost than a new pump.