A moderate, but well built, home theater will be pretty much air tight. All the walls and ceiling (interior walls as well) are insulated. Each person in that room SLEEPING generates 500 BTU/hr of heat (if you're doing something else, more heat). Four people is (tough math problem) is 2000 BTU/hr. What's a small mini split do?
Now, if you live in MiniNoPlace, don't have a isolated, insulated, reasonably air tight room and your electronics are vented outside of the room, perhaps you could get away with no means to cool the room. On the other hand, I have done rooms in MiniNoPlace (and St. Paul), Adak, Colorado (lived there, done that), Ottawa and Vermont and the A/C runs in the dead of winter (even in Vail).
If you run around your house with scissors, you want a very cold house. The cold will reduce the pain, slow the blood flow, reduce your metabolism and give you a better chance of surviving while you're waiting for emergency medical services. That will also mean less blood to clean up from the floor. Ever been diving...in the Aleutians...in February...under the ice pack? That's cold. Your cool basement is a non-issue.