Based on your new pics, I would say the easiest thing to do would be to:
- Turn off the breaker to your thermostat (HVAC system).
- Enlarge the hole slightly to a more even shape.
- Get a piece of wood, not too thick (maybe 1/4"-1/2"), just a bit narrower than the hole, and about 4" longer than the hole is wide. Drill a hole in it for the t-stat wire. If there's enough slack in the wire you can just stick it through the wood. If there isn't enough slack, thread a pull string through the hole and attach it to the wire, so you can pull the wire through later.
- Stick the wood in the hole so that it's flat against the back side of the wall. (It can help to temporarily screw a long screw into the wood near the hole to use as a handle so you can hold the wood in place.)
- Orient the wood so that the ends are behind parts of the wall without existing holes.
- Screw a drywall screw through the wall, into each end of the wood. It may help to drill pilot holes in the wall and wood (you may want a helper to keep the wood in place after drilling the hole, to make sure you hit the hole with the screw). Remove the temporary "handle" screw, if necessary.
- Cut a small piece of drywall, big enough to fit in the hole. Drill a hole in it for the t-stat wire and thread it through. Use construction adhesive or wood glue to glue the piece of drywall to the wood backer. If the hole is really deep, you may need/want two layers of drywall.
- Once the glue is set, you can use standard drying joint compound to fill the holes and gaps. Then sand and paint.
Of course, you want to do these steps when it's relatively cool and don't need the A/C. And it doesn't all need to be done in one go (especially the joint compound part). You can reconnect the t-stat and turn the breaker back on at any point once you're done for the day—just turn off again when you're back working.
Finally, it's not clear if your thermostat wire goes through this big hole, or if it has its own. If the latter, you won't need to drill any holes in your patch for it.