Don't know how well it would work in Phoenix and it depends on the bike but if the seat has rails to adjust the distance from the handlebars a spare innertube can be nicely ziptied up there with little concern of it falling off. But I would definately second the spare tube, besides it can always be filled with water as a canteen as long as you don't Slime the thing. (not sure if they use anything inside or not so I don't know if its safe, just saw it on survirorman)

And I definately second the lights (flashing red on back if possible) and a helmet.