  In this work, we consider 2D $\mathbb{Z}_2$ topologically ordered phases
($\mathbb{Z}_2$ toric code and the modified surface code) on a simple
hyperbolic lattice. Introducing a 2D lattice consisting of the product of a 1D
Cayley tree and a 1D conventional lattice, we investigate two topological
quantities of the $\mathbb{Z}_2$ topologically ordered phases on this lattice:
the ground state degeneracy on a closed surface and the topological
entanglement entropy. We find that both quantities depend on the number of
branches and the generation of the Cayley tree. We attribute these results to a
huge number of superselection sectors of anyons.
