The density of any part of the black hole is constant. (as long as the mass is constant)
The density is not constant, it (mathematically) approaches infinity as you approach the singularity (or center).
At the center of a black hole as described by general relativity lies a gravitational singularity, a region where the spacetime curvature becomes infinite.[52] For a non-rotating black hole this region takes the shape of a single point and for a rotating black hole it is smeared out to form a ring singularity lying in the plane of rotation.[53] In both cases the singular region has zero volume. It can also be shown that the singular region contains all the mass of the black hole solution.[54] The singular region can thus be thought of as having infinite density.
Observers falling into a Schwarzschild black hole (i.e. non-rotating and no charges) cannot avoid being carried into the singularity, once they cross the event horizon. They can prolong the experience by accelerating away to slow their descent, but only up to a point; after attaining a certain ideal velocity, it is best to free fall the rest of the way.[55] When they reach the singularity, they are crushed to infinite density and their mass is added to the total of the black hole. Before that happens, they will have been torn apart by the growing tidal forces in a process sometimes referred to as spaghettification or the noodle effect.[56]
In the case of a charged (Reissner–Nordström) or rotating (Kerr) black hole it is possible to avoid the singularity. Extending these solutions as far as possible reveals the hypothetical possibility of exiting the black hole into a different spacetime with the black hole acting as a wormhole.[57] The possibility of traveling to another universe is however only theoretical, since any perturbation will destroy this possibility.[58] It also appears to be possible to follow closed timelike curves (going back to one's own past) around the Kerr singularity, which lead to problems with causality like the grandfather paradox.[59] It is expected that none of these peculiar effects would survive in a proper quantum mechanical treatment of rotating and charged black holes.