Relatively speaking, forcible detainer actions--eviction lawsuits--in Arizona are a quick and inexpensive way to obtain a judgment against a tenant. If an Arizona tenant vacates the property prior to the eviction hearing then we are not able to continue to with the lawsuit. You can't try and evict someone when you know they are no longer living in the property.
However, even if the tenant has vacated the property there is still an available remedy to try and seek a judgment against your former tenant. In this situation you may sue the former tenant under a breach of contract. The theory behind this is, a lawsuit existed (the lease agreement) and the contract was broken.
Unlike an eviction lawsuit a civil lawsuit is very slow moving and the attorneys' fees are much more expensive. Depending on many factors; whether the tenants counter-sue, hire legal counsel, are difficult to serve, and an extensive amount of discovery is required the typical case can range between 6 to 16 months! Much longer than the average 21 - 28 days required for a residential eviction.
In 99% of the instances it does not make economic sense to try and sue your former tenant through a breach of contract with a civil lawsuit. It's just not worth the time, expense, and hassle of trying to get a judgment against them. Lastly, don't forget, just because you actually obtain a judgment against your former tenant doesn't mean that you'll be able to collect on it! Often the judgment isn't worth the paper it's printed on.