I upgraded my thermostat a couple years ago and had the same issue, not realizing that I was missing a "C" / Common wire to provide continuous 24V to the thermostat, the batteries were really just for backup to save schedules,settings and whatnot in case of power loss.
I just installed a Sensi wifi thermostat at home. So long as you have 24v continuous power through the c-wire the batteries shouldn't be used I would think. My guess is a corroded or loose connection on your c-wire. I had to run a new one from my furnace - took 10 minutes and about $10 worth of wire.
If the batteries are draining quickly I'd assume its some sort of problem though. You said you don't own, did you contact the landlord?