Is your attic well insulated? That will DEFINITELY help if it is not.. or poorly insulated.
Another thing is attic ventilation. By installing a gable end vent fan which cycles on temperature, you can lower the attic temperature drastically which is adding a heat load to your upstairs rooms.
Other ideas.. yes closing the doors will probably help if there is a return in each room. How many thermostats/zones are there. If there is only one, and the downstairs is cool and the upstairs hot, you can closedown a little on the downstairs vents to get more air upstairs, and it will allow the thermostat to see a more representative sample. Further, leaving the upstairs doors closed will prevent the cold air from up there from falling to the downstairs cooling it off further causing the ac to cycle off prematurely.