You posted under Aube in-wall timer.
If this is not your timer, add a comment.
Aube timers have a rechargeable battery that runs timer functions, and timer does not require 120 volts from circuit.
OK, I wrote to Aube about low temperatures affecting their battery.
Aube said, the battery is good down to freezing, and probably lower.
Lower temperatures cause display to slow down and not function, which means timer programs will not work when timer is exposed to cold temperatures.
Manual override would still work, but not programming or display.
So if timer is NOT located inside warm house, for example timer located in unheated garage or shed, then timer will not function in cold temperatures.
That being said, Aube battery is rechargeable.
Battery is not replaceable.
If Aube battery has gone bad, then timer is not repairable.
Intermatic ST01C and EJ500 in-wall timers are also battery operated. Intermatic ST01C and EJ500 battery can be replaced.
Almost all plug-in type timers also have a rechargeable battery that is not replaceable.
Several in-wall timers do not require a battery, but they need power from electric circuit.
To see these timers, plus description how to wire this type of timer:
http://waterheatertimer.org/Programmable.html#Aube