It's fine. There won't be any problems.. A lot of people actually pick their oil due to the regional temperature they live and operate the vehicle in.
Viscosity (a fluid's resistance to flow) is rated at 0° F (represented by the number preceding the "W" [for Winter]) and at 212° F (represented by the second number in the viscosity designation). So 10W-30 oil has less viscosity when cold and hot than does 20W-50. Motor oil thins as it heats and thickens as it cools. So, with the right additives to help it resist thinning too much, an oil can be rated for one viscosity when cold, another when hot. The more resistant it is to thinning, the higher the second number (10W-40 versus 10W-30, for example) and that's good. Within reason, thicker oil generally seals better and maintains a better film of lubrication between moving parts.
At the low-temperature end, oil has to be resistant to thickening so that it flows more easily to all the moving parts in your engine. Also, if the oil is too thick the engine requires more energy to turn the crankshaft, which is partly submerged in a bath of oil. Excessive thickness can make it harder to start the engine, which reduces fuel economy. A 5W oil is typically what's recommended for winter use. However, synthetic oils can be formulated to flow even more easily when cold, so they are able to pass tests that meet the 0W rating.
Once the engine is running, the oil heats up. The second number in the viscosity rating--the "40" in 10W-40, for example--tells you that the oil will stay thicker at high temperatures than one with a lower second number--the "30" in 10W-30, for example.