We study the theory of Jackiw-Teitelboim gravity with generalized dilaton potential on Euclidean two-dimensional negatively curved backgrounds. The effect of the generalized dilaton potential is to induce a conical defect on the two-dimensional manifold. We show that this theory can be written as the ordinary quantum mechanics of a charged particle on a hyperbolic disk in the presence of a constant background magnetic field plus a pure gauge Aharonov-Bohm field. This picture allows us to exactly calculate the wavefunctions and propagators of the corresponding gravitational dynamics. With this method we are also able to reproduce the gravitational density of states as well as compute the R\'eyni and entanglement entropies for the Hartle-Hawking state. While we reproduce the classical entropy at high temperature, we also find an extra topological contribution that becomes dominant at low temperatures. We finally show that these quantities are also obtanable by dimensionally reducing $SL(2,\mathbb{Z})$ black holes of pure 3D gravity.
