  In this paper, we generalize our results in \cite{GX3} to triangulated
surfaces in hyperbolic background geometry, which means that all triangles can
be embedded in the standard hyperbolic space. We introduce a new discrete
Gaussian curvature by dividing the classical discrete Gauss curvature by an
area element, which could be taken as the area of the hyperbolic disk packed at
each vertex. We prove that the corresponding discrete Ricci flow converges if
and only if there exists a circle packing metric with zero curvature. We also
prove that the flow converges if the initial curvatures are all negative. Note
that, this result does not require the existence of zero curvature metric or
Thurston's combinatorial-topological condition. We further generalize the
definition of combinatorial curvature to any given area element and prove the
equivalence between the existence of zero curvature metric and the convergence
of the corresponding flow.
