Eight, it is rooms, not doors.
The instructions could certainly be clearer here though.
Many room types provide "connected" points if entrances from the current room are connected to entrances of rooms of specific types
There is nothing there that says that you score only once per room, or really even that you score more if connected to more than one room of a given type.
Its seems intuitive that it would be once per room, but after several plays someone noted that the rule states that the bonuses are scored when entrances are connected. It did cause us to question our original reading because the case can be made that you get points for each properly connected entrance, and not for each room.