It uses the battery from whatever device it's plugged into via usb-c. I haven't noticed a big battery impact difference between the on board screen and the glasses. I have my ROG allyX (running bazzite) set up to turn off the on board screen when the glasses are plugged in.
They do get warm during long sessions, but not uncomfortable. Especially when compared to holding the ally up and looking down at it. You can rest it on your lap and look in any direction.