(Dokuchaev 2011) found periodic orbits of particles and photons inside a Kerr-Newman black hole with a maximally extended global geometry. They orbit inside the inner horizon $r<r_-$ where the $r$ coordinate becomes space-like again. Leaving stability and habitability aside, how much space is there in this region to orbit?
The internal volume of black holes is somewhat tricky to define. (Christodoulou and Rovelli 2015) worked out one approach that gives a large, but finite, interior volume of black holes. Volume is defined as the volume of the largest spacelike spherically-symmetric surface that has the 2-sphere of the black hole at some point in time as boundary. (Bengtson and Jakobsson 2015) have extended the approach to rotating black holes. However, it is not clear to me how to apply their method to determining how much volume there is inside the inner event horizon.
A naive calculation based on the determinant (107) in (Visser 2008) seems to imply that the spatial volume should be $$V=\int_{r=0}^{r=r_-}\int_{\theta=0}^{\theta=\pi}\int_{\phi=0}^{\phi=2\pi} \sqrt{\sin^2\theta (r^2+a^2\cos^2\theta)(r^2-2mr+a^2)}dr d\theta d\phi$$ but when I try to work through the integral for $\theta$ I end up with a logarithmic singularity as $r\rightarrow 0$. Now, given that there are other domains in the extended geometry that might not be entirely odd, but I was assuming this integral only deals with the interior region rather rather than any of the weird extensions. On the other hand, using equation (9) from Bengtson and Jakobsson seems entirely straightforward and gives a similar-looking formula without the singularity.
I assume the Christodoulou and Rovelli approach would start by considering the $r=r_-$ surface and for a given moment of coordinate time try to find the largest spacelike axially symmetric 3-surface that has that as a boundary.
