  We apply Selberg's trace formula to solve problems in hyperbolic band theory,
a recently developed extension of Bloch theory to model band structures on
experimentally realized hyperbolic lattices. For this purpose we incorporate
the higher-dimensional crystal momentum into the trace formula and evaluate the
summation for periodic orbits on the Bolza surface of genus two. We apply the
technique to compute partition functions on the Bolza surface and propose an
approximate relation between the lowest bands on the Bolza surface and on the
$\{8,3\}$ hyperbolic lattice. We discuss the role of automorphism symmetry and
its manifestation in the trace formula.
