Engine temperature gauge reads cold

Cold can drain battery power especially if the battery is old. How cold is it and how oil is your battery?

If the gauge always reads cold it could be a stuck open thermostat (no heat is another symptom of that), bad coolant temperature sensor (check engine light should be on), bad wiring going to the coolant temperature sensor (check engine light on), or a bad ground for the gauge. But it probably won't kill the battery. Hope that helps! Jim