  Motivated by the recent experimental realizations of hyperbolic lattices in
circuit quantum electrodynamics and the research interest in the non-Euclidean
generalization of topological phenomena, we investigate the Chern insulator
phases in a hyperbolic $\{8,3\}$ lattice, which is made from regular octagons
($8$-gons) such that the coordination number of each lattice site is $3$. Based
on the conformal projection of the hyperbolic lattice into the Euclidean plane,
i.e., the Poincar\'{e} disk model, by calculating the Bott index ($B$) and the
two-terminal conductance, we reveal two Chern insulator phases (with $B=1$ and
$B=-1$, respectively) accompanied with quantized conductance plateaus in the
hyperbolic $\{8,3\}$ lattice. The numerical calculation results of the
nonequilibrium local current distribution further confirm that the quantized
conductance plateau originates from the chiral edge states and the two Chern
insulator phases exhibit opposite chirality. Moreover, we explore the effect of
disorder on topological phases in the hyperbolic lattice. It is demonstrated
that the chiral edge states of Chern insulators are robust against weak
disorder in the hyperbolic lattice. More fascinating is the discovery of
disorder-induced topological non-trivial phases exhibiting chiral edge states
in the hyperbolic lattice, realizing a non-Euclidean analog of topological
Anderson insulator. Our work provides a route for the exploration of
topological non-trivial states in hyperbolic geometric systems.
