There isn’t a best time to build a new pool per se, but there are a couple things to consider. For starters if you want to swim when summer arrives, you should start the process several months in advance. The permit process could range from one day to several months depending on the timing, municipality and city staff workload. New pool construction will generally take about 2 months depending on the size and complexity of the pool or spa.
On the other hand, if you don’t live in Arizona permanently more than likely you are gone in the summertime so this might be the most ideal time for pool construction, with proper planning the entire pool and spa, as well as landscaping, can be done which you are away so it is ready for you to enjoy upon your arrival.