Spinning black hole laser? Is a laser that uses a rotating black hole's ergosphere as the gain medium possible?  The idea is that the coherent light feeds off of the black hole's rotation.  Assume that the black hole is rotating at a substantial fraction of $c$, as is realistic.
Edit: The original context is that a really advanced civilization that orbits a black hole wants to extract energy from the BH without having to send something valuable (like a spaceship) near the BH, or wants to create an extremely powerful directed energy weapon to defend its system against hostile attackers.  You can make really big mirrors in space, so this seemed like a useful concept.
 A: Blue shift is something that occurs both as you observe a light (reflected or emitted) source that is approaching you or as light descends toward you down a gravitational potential. These work together in the scenario you put forward, as explained in this Kurzgesagt video. This is not dependent on coherent light though.
The advantage of using coherent light is that it can interfere with itself within an overlapping ring or reflection in an optical cavity. In terrestrial cases, a gain medium is placed in the path of the beam and excited by an external power source to great effects.

Optical ring resonator
Perhaps there is an a way to position terrestrial mirrors such that a laser can steal momentum from the ergosphere, but you would need reflective material that could withstand a .001% energy conversion to heat on each reflection.
With current materials, you might be better off directing the laser in a path that is almost within the photon sphere of the black hole, before redirecting it to be harvested with a disposable mirror. As beam divergence occurs, there will be trade-offs in how many mirrors or lenses you can sacrifice to make this beam go where it needs to.
It seems possible to use a black hole's ergosphere as a gain medium.
