It seems to take the approach of a post-processing frame shift rather than real-time stabilisation. In other words it lets the phone capture and compress using hardware, but keeps the sensor data separately and shifts the x,y of the video to stabilise. Unlike Final Cut etc. that has to analyse each frame, Luma has the raw data as the video was being shot. So the battery hit would come on export when the video has to be re-rendered I'm guessing.
I think this is the method because the Luma video appears more zoomed in than the original, i.e. they leave some space around so the video is always visible. This means that the Luma output will be lower res than the original hardware captured video I'm guessing.
Still, if it avoids walking around with your phone in a steadicam rig then it's a huge win. I have a steadicam smoothee but the ridicule from friends and family mean I don't use it, the slightly lower res output with Luma is the price my family will pay for their insensitive mockery!