Grand Canyon University was originally founded in Prescott, AZ in 1949. Shortly thereafter the campus was moved to Phoenix in the early 1950's. The mountains and high desert around Prescott hold a good population of antelope, hence the mascot. Why the school's founding fathers chose the antelope rather than the more intimidating scorpions, killer spiders, or mountain lions, which also inhabit the area, is beyond my knowledge.
Side note: The Pronghorn 'antelope' which inhabit the area around Prescott are not even antelope at all. Rather, they are the only member of the family antilocapridae. There are actually no true antelopes in North America, and the Pronghorn, as it should be called, is most closely related to the Giraffe.