The only time you're going to really notice a big drop in performance is if you've got cold wax on and it's spring-like, wet and slushy conditions. Then you'll notice the base/water suction slowing you down in the flats and cat-tracks. Hydrophobic (is that the term?) warm wax repels water and helps break that suction. Frankly, there probably won't be enough melting on a mild January day to create much water.
If your board needs a waxing anyway, then go ahead and put the right temp range wax on. If it doesn't need a waxing, then don't bother