Are there any solutions to Einstein's equations which are periodic in time?
A black hole only has mass, charge and angular momentum according to the no-hair theorem. (Although this might just mean in the limit as time goes to infinity because when two black holes collide the shape of their combined horizon would be extra information).
Would a binary black hole system be periodic? I'm not sure because it might give off gravitational waves and collapse. (But there is also the time reverse of this where a single black hole absorbs gravitational waves and splits in two!)
What about a pair of black holes in a Universe that is expanding so fast that they always stay the same distance apart?