iPhone claiming low battery in cold weather

I would question two things about your framing of the issue (and will sidestep the whole repair discussion/issue entirely):

1) Apple's warranty does cover batteries that don't retain a percentage of their useful life based on aging at one, two and three years (two for most iOS warranty / coverage, but there are exceptions in some cases). You should open a ticket with them if you have coverage and see if they will repair it for you.

2) I use an iPhone regularly outdoors when the temperature reaches -30 C and even then, the decrease in capacity is temporary. While the device is colder than 0C - you get less juice out of the cell, but once you warm it (in a pocket, next to your skin to warm it temporarily - or when you are inside a heated space for long enough to warm the battery), the devices recover full capacity.

(You will want to watch to control condensation to prevent liquid damage / especially if you bring a very cold device into a warm humid space, that warm air will carry liquid inside the product. Similarly, a warm device when rapidly cooled could have liquid condense inside it and then freeze.) Devices left in a car overnight such as iPad and Mac also see no measurable decrease in battery life once I get them inside and warm - whether I charge them or not.

I'll leave the DIY repair / pay for a repair discussion for another thread where you get specific on what model, but from what I see, the environmental behavior is that cold batteries still have all the charge they took, but they don't produce as much voltage and current while they are cold.

The operating temperature of an iPhone 4s is 0 to 35 degrees Celsius (source: http://store.apple.com/us/browse/home/shop_iphone/family/iphone4s). At minus 12 degrees C, you're below the threshold so your phone behavior is expected.

To keep iPhone warm in cold environment, I put it in the inner vest or jacket pocket as that will be warmer with the insulation and proximity with the body. So far no problem from phone not working. The main problem is my gloves prevent my fingers from touching the glass surface resulting in unintended app actions.