If the starter is turning over normally, then the battery isn't the culprit here. At minus five, the problem is usually in the fuel system or spark plugs. The first thing to check is the condition of your spark plugs. They often misfire in the cold, especially if they haven't been changed in a while. Try unscrewing them and looking at the gap and carbon buildup, or just replace them with new ones - it's cheap and often does the trick.
Second - fuel filter and pump. If the filter is clogged, fuel might not be getting through in the right amount, and you'll notice that right away in the cold. The pump can also act up on frost, though a new battery rules out a complete electrical failure. Also keep in mind that fuel thickens in cold, and if yours is low quality, it might not flow properly through the system.
One more thing - check if there's an ignition issue. Try turning the key and sitting for about 10-15 seconds to give the pump time to prime the fuel, then start it. If that helps, the problem's definitely in the fuel system. If nothing changes, get it to a diagnostics shop - they'll read the error codes from your car's computer, and it'll become clear pretty quick what's wrong.