What stops giant cruise ships toppling over in rough seas?

Some dimensions I was able to dig up (mostly from Wikipedia).

Draft of the Allure of the Seas: 31 ft (10 m)
Length: 1181 ft (360 m)
Beam at waterline: 47 m Height: 72 m above waterline

Let's just draw the section based on these simple numbers:

Now if the center of gravity were in the middle of the ship (31 m above the water line), it would indeed not be very stable - any tilt beyond 25° would cause it to tip over:

However, there are several important factors:

1. The part of the hull below the surface is made of much thicker, stronger, heavier material than the superstructure
2. The engines etc. are all in the lowest levels
3. There is an active ballast system that allows pumping fuel and sea water from side to side to help maintain balance

I found one online test of the stability of the hull of this ship in a test facility, where they made a big hole in the side of an accurate model to see how it would fare: https://www.youtube.com/watch?v=Ra4TkHOs4RE . While there are commercial interests at stake, nobody wants a marine disaster on their hands.

I doubt it is all fluid dynamics. The have to stay upright even with dead engines. If the integral of the lever below the water line is bigger than above then it should stay upright. Ballast at the bottom goes a long way as it has a long lever. Stuff like engines below deck tends to be heavy anyway. Weight is not a big deal as they are not going up hill. They need a low draft to get into port. If you looked at an aircraft carrier out of water it would also appear top heavy.

Let assume a ship with 2/3 above water line. Break the up in 9 units. 6 above and 3 below.
Naturally lightest stuff on top and heavy below. In this example the net lever above is 45 and below 180.

Height Weight Lever 
6      1       6
5      2      10
4      3      12
3      4      12
2      5      10
1      5       6
net           56
1      10     20
2      20     40
3      40    120
net          180

Imagine that is a teeter-totter with 3 fat kids on one side and 6 skinny on the other. Even if the skinny kids have twice the lever the fat kids still dominate.

Since it pivots on the hull and not the center line the bottom loses leverage as it starts to list. So a narrow hull at the water line is a good thing.