In AZ Open carry is legal and you might get challenged on it in the city as to it being needed and many businesses do not allow open carry or even conceal carry if the sign is of any indication. There are places it is illegal which make sense, schools, gov't buildings, hospitals, theaters, etc.
But, you cannot legally ban the weapon from being in the vehicle locked away. Having it in the vehicle not in a gun safe bolted to the vehicle is reckless and IMHO pretty stupid. Small gun safes can be bolted down in the trunk or center console that will defeat at least temporarily all but the determined thief.
Even concealed carry is allowed there without a license, unless the laws were changed since spring of '13 when I moved back to TN after nearly a 30 year absence.
Castle law is in effect in AZ as in TX and when at home you have to right to shoot anyone who you feel is threatening your life with bodily harm or others are about to receive same harm and they are unable to protect themselves in your home. Basic gist of the castle doctrine law...
David