I might be wrong on this, but my impression is that the gravitational slingshot relies on the fact that the slingshotting planet is orbiting the Sun and you pass behind it and have it pull you forwards at the expense of its own orbital angular momentum. This means that if you have a stationary black hole, it's no better for slingshotting than the Sun is...
That being said, if you have a rotating black hole, which is the case for just about every astronomical black hole, there's a region that is outside the event horizon but where spacetime gets dragged faster the speed of light in the direction of rotation, called the ergosphere. Since it's outside the event horizon there's no problem in entering and exiting this region, and if you're clever about how you do it, you can actually leave with more energy than you came in. This is called the Penrose process
Our universe is most certainly unique... it's the only one that string theory doesn't describe.