  When working with asymptotically hyperbolic initial data sets for general
relativity it is convenient to assume certain simplifying properties. We prove
that the subset of initial data sets with such properties is dense in the set
of physically reasonable asymptotically hyperbolic initial data sets. More
specifically, we show that an asymptotically hyperbolic initial data set with
non-negative local energy density can be approximated by an initial data set
with strictly positive local energy density and a simple structure at infinity,
while changing the mass arbitrarily little. This is achieved by suitably
modifying the argument used by Eichmair, Huang, Lee and Schoen in the
asymptotically Euclidean case.
