  In this paper we show that any good toric contact manifold has well defined
cylindrical contact homology and describe how it can be combinatorially
computed from the associated moment cone. As an application we compute the
cylindrical contact homology of a particularly nice family of examples that
appear in the work of Gauntlett-Martelli-Sparks-Waldram on Sasaki-Einstein
metrics. We show in particular that these give rise to a new infinite family of
non-equivalent contact structures on $S^2 \times S^{3}$ in the unique homotopy
class of almost contact structures with vanishing first Chern class.
