There was racing in Phoenix up until about 10 years ago, and several farms that raised and trained them here, so yes, you can certainly have greyhounds in Phoenix. However, they are pretty heat sensitive, and you have to be careful not to let them get overheated in the summer, or really any time of year. As Don said, walk early in the morning, stay on grass or dirt as much as possible, keep walks short and in the shade as much as possible. I have noticed that as they get older the heat seems to bother them more.
Some people use cool coats although I never have. Many of mine have enjoyed laying out in the sun in 100+ temperatures I just go out and make them come in after a few minutes