How do battery powered thermostats with LCD screens run for so long?
LCDs (liquid crystal displays) require very little power to operate.  The liquid crystal looks mostly like a small capacitor.  To keep the liquid crystal in the dark state, a few volts of AC at a few kHz or so must be applied across this capacitor.  That's not a lot of power.

This is exactly why you see LCD readouts and displays in low power battery operated devices to often.  Think of a wristwatch.  Those run for a year or more on a tiny battery.