How is battery life measured in devices?
Typically low battery (for primary- aka non-rechargeable) batteries or cells is indicated when the voltage (preferably under load) drops below a set threshold. 

Replacing primary alkaline cells with a different type such as NiMH rechargeable cells can result in a 'low battery' indication because the cell chemistry is quite different, resulting in a lower voltage- 1.2V vs. 1.5V in that particular case- even for freshly charged batteries.  

We have no way of knowing whether attaching an external power supply could cause any damage- if the thermostat has problems replacing it or repairing it would probably be a better course of action.