  We investigate the Holographic Entanglement Entropy proposal in the context
of the (3+1)-dimensional topological black hole. In contrast to the
well-studied (2+1)-dimensional case, the maximal extension for this black hole
includes only a single exterior region with its conformal boundary. This
immediately raises a puzzle as to how one can view the purification of the dual
conformal field theory state in terms of a thermofield double in the usual
manner. Motivated by this puzzle, we calculate the horizon area for these black
holes and discover that the result is observer dependent. This observer
dependence poses a potential issue in applying the holographic entropy
proposal. Investigating this we find that, although this observer dependence
does not carry over to the holographic entanglement entropy, there is an
indication of a coordinate system which is best adapted for the holographic
calculation. These coordinates only cover two regions of the spacetime which
exactly correspond to the regions of the CFT on which particle modes are well
defined and so we see that the holographic calculation in the spacetime is
capable of predicting regions of the CFT where particles cannot exist.
