Check the update settings if that feature. It could be set to update at certain intervals which are longer than you'd like. Remember, the frequent the update the more battery use will be required to fetch the most current information.
And some weather apps have the choice of showing your home location or showing your current location. Make sure yours is set to current location if it has that setting. (Also remember than temperature can vary a few degrees in less than a mile. Even the difference between being under a cloud and being in direct sunlight can give you a 5 degree difference in still air over a distance of 10 feet.)