I'm becoming sad as I see post after post treating it as "obvious" that it requires continuous power input to hover. It really, really, truly, does not. Energy is required to rise in a potential field (like a gravity well), and energy is dissipated when you fall in a potential field. If you are stationary, your potential energy is constant, and thus no energy is needed to maintain your position.
This is entirely distinct from the question of force. An object in a potential field experiences a force in the direction of lower potential, and if nothing counters that force, it accelerates in that direction. For a hovering helicopter, the counterforce is provided by pushing a lot of air downward. For my phone right now, which is "hovering" a meter above the floor and ~6000 km above the center of mass of the Earth, the counterforce is provided by the surface of the table it's resting on. One requires a lot of continuous power input; the other requires no power input at all.
If you could somehow (handwave, handwave) shield an object from the force induced by a potential field, you could hover in that field at the energy cost of whatever it takes to maintain that shield. Full stop.