Not sure of your mechanical ability, but the first thing to find out is WHAT is making the humming? This requires hands-on time, but novice level stuff.
The top cover and probably an insulation piece (or several...) should be removable without major issues. This may allow you to track down the hum, and/or determine what is NOT humming. Would be nice to know its not the compressor, fan or other $$$ piece.

If you can, post the make/model of your AC/heat pump.
Proper diagnostics begin with good baseline info. The hum might be as innocent as a 120V-24V transformer that is a bit loose.

That would be an item that could be on all the time independently of the thermostat status. There may be gas solenoid(s) that are in a 'locked on' state when the unit is on standby that are vibrating. OR a latching relay that isnt quite latching.
My RV has the old 'resistor-style heat strip' not a heat pump. No wall thermostat control, just the knobs on the unit itself. I would have to see a schematic of your heat pump unit to see if there are any other likely suspects for a 'power-on hum' beyond these guesses.