Power management was initially in its default state with some kind of screensaver and then the device was off after some time. I changed it today to no power saving, but I don't think it made a difference. I am not sure what use is the screensaver - as far as I know e-ink devices consume energy only when the page is changed and turning on and off the device (whether automatically or manually) could use more energy.
Wifi is automatically turned on and off when the browser is open, but I didn't leave the device with the browser open (I was reading a book).
While the pen was unresponsive the battery indicator at the low right corner was crossed out, but I was still capable of changing pages with the buttons (maybe touch screen requires more energy?). But I still don't know if touchscreen is related to energy consumption/batery leak - it is just a suggestion.