typically they let you use outside air when the temp is suitable . . . you need a fresh air intake duct with a controllable damper (not usually found in residential, but required in new commercial work (in Metro-NY area at least)
I'm not sure what the economizer is, or does. .
. . . ie: on a cool evening you would just bring cool fresh air in for your overheating home theater . . .
. . . or another possibility . . .
. . . have you considered installing an exhaust fan and air transfer for your home theater . . . the exhaust fan would be controlled by a thermostat . . . when the t-stat says it's too warm, the fan runs, pulling hot air from the space (and dumping it somewhere else in the upstairs zone), and the transfer (just a hole in the wall with grills both sides, preferably the opposite corner of the room, down low to the floor) lets cool air in . . . the fan should be mounted away from (and ducted to) the HT to avoid the noise . . .
I love my country, but fear my government.