  We establish a connection between the electromagnetic Hall response and band
topological invariants in hyperbolic Chern insulators by deriving a hyperbolic
analog of the Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula. By
generalizing the Kubo formula to hyperbolic lattices, we show that the Hall
conductivity is quantized to $-e^2C_{ij}/h$, where $C_{ij}$ is the first Chern
number. Through a flux-threading argument, we provide an interpretation of the
Chern number as a topological invariant in hyperbolic band theory. We
demonstrate that, although it receives contributions from both Abelian and
non-Abelian Bloch states, the Chern number can be calculated solely from
Abelian states, resulting in a tremendous simplification of the topological
band theory. Finally, we verify our results numerically by computing various
Chern numbers in the hyperbolic Haldane model.
