Why cold weather kills your doorbell battery

A battery that lasted two months in September lasts two weeks in January. Nothing has broken. The chemistry just behaves differently in the cold, and one consequence of that catches almost everybody out.

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A doorbell on exterior trim with frost on the siding around it
Nothing has broken. Battery chemistry simply behaves differently once the wall is this cold.
Why a hardwired doorbell still dies

This is why a hardwired doorbell can still die in winter. The wiring is fine. The camera is refusing the charge on purpose, and it is right to.

Because the wiring is not the problem. Lithium cells should not be charged below freezing and the camera refuses the charge on purpose to avoid permanent damage, so a trickle charge that works all summer simply stops once the temperature drops. The wiring is fine and the camera is right.
Around freezing, as a rule of chemistry that holds across models, and manufacturers set their own cut-off a little above or below it. Check your own model's manual for the figure, because it varies by model.
Only at the margins. Keeping wind and driven rain off the housing is worth something, but no cover keeps a cell above freezing through a hard winter, because the camera generates almost no heat of its own. Buy one for the rain, not for the battery.
Usually not. Capacity lost to cold comes back when the cell warms up, and a battery that recovers with the weather was never faulty. A cell that stays poor in mild weather has aged instead, and batteries are consumable parts.
In a cold snap it is the single most effective thing you can do, and it is also the most tedious. Where the battery is removable, a charged spare turns a day of downtime into a two-minute swap, which is the standard answer in cold climates.

Wiring it, if you can

Dual-lens wired doorbell outside a door with its chime plugged in indoors
Wired

eufy Video Doorbell S330, wired, dual camera

A wired dual-camera unit with a second lens aimed at the step. That is the fix for the parcel blind spot a single wide lens handles badly. Check the wiring before anything else. It wants 16 to 24 volts at 30VA. That is well above what an older bell transformer supplies, so a healthy 10VA unit still will not run it. A chime is included, and it does not work with a eufy HomeBase.

Battery glyph for Ring Quick Release Battery Pack
Battery

Ring Quick Release Battery Pack

The spare the cold-weather page tells you to keep. A second pack turns a day of downtime into a two-minute swap. It also charges indoors at room temperature, which is the thing that actually helps in winter. Check the listing's model list before buying: it names a long run of Ring cameras and doorbells, and yours has to be on it. The quick-release tab means the camera stays on the wall while you change the pack.

Weather glyph for Acrylic doorbell rain and sun cover
Weather

Acrylic doorbell rain and sun cover

Read what this does and does not do before you buy it. A hood keeps driving rain and low sun off the lens, and it keeps wind off the body. It will not keep a battery above freezing in a hard winter, and anyone selling it as a cold-weather fix is overselling it. Acrylic, 3.9 inches wide, and it goes on with screws or with the double-sided tape in the box, so it suits a rented door as well.