  We formulate the most general gravitational models with constant negative
curvature ("hyperbolic gravity") on an arbitrary orientable two-dimensional
surface of genus $g$ with $b$ circle boundaries in terms of a
$\text{PSL}(2,\mathbb R)_\partial$ gauge theory of flat connections. This
includes the usual JT gravity with Dirichlet boundary conditions for the
dilaton field as a special case. A key ingredient is to realize that the
correct gauge group is not the full $\text{PSL}(2,\mathbb R)$, but a subgroup
$\text{PSL}(2,\mathbb R)_{\partial}$ of gauge transformations that go to
$\text{U}(1)$ local rotations on the boundary. We find four possible classes of
boundary conditions, with associated boundary terms, that can be applied to
each boundary component independently. Class I has five inequivalent variants,
corresponding to geodesic boundaries of fixed length, cusps, conical defects of
fixed angle or large cylinder-shaped asymptotic regions with boundaries of
fixed lengths and extrinsic curvatures one or greater than one. Class II
precisely reproduces the usual JT gravity. In particular, the crucial extrinsic
curvature boundary term of the usual second order formulation is automatically
generated by the gauge theory boundary term. Class III is a more exotic
possibility for which the integrated extrinsic curvature is fixed on the
boundary. Class IV is the Legendre transform of class II; the constraint of
fixed length is replaced by a boundary cosmological constant term.
