It seems to be the screen. When I switch off the option to let the screen on the screen gets indeed dark after the typical timespan. When I let it this way (with the camera open) it seems not to drain the battery at all. The device does also not heat up anymore.
The auto focus seems not to be a significant part of the problem. I tried also to switch it off by choosing one of the several manual focus methods, but this didn’t show any effect on the battery. I also can’t detect an impact on CPU load.
BTW: it seems that a manual focus is much more difficult to use than just to rely on the auto focus, especially when you don’t use a stand. I got not many good pictures using a manual focus, even when I tried very carefully to focus something. Especially on near distances (under 1m) the continuous auto focus is much better. So I will not switch it off generally.