A double can be completed on just about any bike you can endure sitting on for the duration. More importantly you need to have the engine ready. I have completed numerous centuries and double metrics this season - but I have yet to bite the bullet and go for 200mi in a day. It is quite a commitment. Without knowing a lot about your capability, I would guess that it would take you anywhere from 13 to 16 hours of saddle time to complete on a flat route.
How far are you riding currently?
I would recommend the same bike as I would if you were riding a metric century, century, double metric or double. Choose a bike that comfortable yet efficient. My priorities would be to have strong wheels, low rolling resistance tires, the option for a wind cheating position (not completely upright), wide gearing range, varied hand positions and some light cargo carrying capacity (a rear rack).
At the slower but potentially more comfortable end of the spectrum are hybrid / fitness bikes. They are efficient enough to get the job done and would not be a big change in riding style from your mountain bike.
To go a bit faster, cyclocross, touring or endurance bikes fit the bill. These will have a nice balance of features, are pretty robust and will be much more efficient than a hybrid. The riding position is more agressive than a hybrid or MTB, but likely still comfortable.
A road bike with a race geometry will be a touch faster yet, but it comes at the cost of a more aggressive position which may sacrifice comfort in the long run.
TT/Tri bike - fast but, not recommended for an enjoyable ride.