It isn't the stove but the fuel that is a problem at low temps. Be sure your fuel canister contains a mix of isobutane and propane. Butane (as opposed to isobutane) won't vaporize below freezing, and IMHO gets sluggish below about 40*F. Isobutane gets sluggish around 20*. In either case, it will let the propane burn off first, leaving you with an unresponsive stove.

The canister needs to be warm when you start it and needs to be kept warm during use (evaporation of fuel while in use chills the canister back down below freezing in a big hurry). You can put the canister in your armpit under your insulating jacket and dance around (it feels like a lump of ice--it will really wake you up in the morning!) for about 15 minutes. For the morning, you can keep the canister in the bottom of your sleeping bag overnight along with your water filter, camera and other items with batteries, causing lots of clunking every time you turn over during the night. I haven't tried the pan of water (requires carrying an extra pan), but I do set the canister on a circle cut from an old blue foam pad. I use a windscreen to keep the canister warm while the stove is burning and for more efficient use of fuel. IMPORTANT SAFETY NOTE: Canister stove manufacturers say not to use a windscreen. Translation, be very careful if you do! Feel the canister frequently and don't let it get hot. An exploding canister will ruin your whole trip, if not worse! The windscreen I use goes only 7/8 of the way around the canister when my pot is on top.

I've used the above methods (starting with warm canister and using the blue foam and windscreen while the stove is burning) at 15*F (at 4900 ft. elevation) and it worked fine. Much colder than that, you should invest in a remote canister stove that allows you to use the canister upside down (that solves the problem of the propane burning off first). Those can be used around 0*F or below, so I've read (I don't camp when it's that cold, thank you!). Or buy a white gas powered stove, put up with the extra weight of stove, fuel bottle and fuel, and get warm pumping it instead of dancing with icy canister in your armpit.

Be sure to warm up your butane lighter before trying to use it, too! If it gets cold enough, it may not light even if prewarmed. Take matches as a backup.