  We study the subelliptic heat kernel of the sub-Laplacian on a
2n+1-dimensional anti-de Sitter space H2n+1 which also appears as a model space
of a CR Sasakian manifold with constant negative sectional curvature. In
particular we obtain an explicit and geometrically meaningful formula for the
subelliptic heat kernel. The key idea is to work in a set of coordinates that
reflects the symmetry coming from the Hopf fibration S1->H2n+1. A direct
application is obtaining small time asymptotics of the subelliptic heat kernel.
Also we derive an explicit formula for the sub-Riemannian distance on H2n+1
