  We analyze the thermodynamic properties of the random-bond Ising model (RBIM)
on closed hyperbolic surfaces using Monte Carlo and high-temperature series
expansion techniques. We also analyze the dual-RBIM, that is the model that in
the absence of disorder is related to the RBIM via the Kramers-Wannier duality.
Even on self-dual lattices this model is different from the RBIM, unlike in the
euclidean case. We explain this anomaly by a careful re-derivation of the
Kramers--Wannier duality. For the (dual-)RBIM, we compute the
paramagnet-to-ferromagnet phase transition as a function of both temperature
$T$ and the fraction of antiferromagnetic bonds $p$. We find that as
temperature is decreased in the RBIM, the paramagnet gives way to either a
ferromagnet or a spin-glass phase via a second-order transition compatible with
mean-field behavior. In contrast, the dual-RBIM undergoes a strongly first
order transition from the paramagnet to the ferromagnet both in the absence of
disorder and along the Nishimori line. We study both transitions for a variety
of hyperbolic tessellations and comment on the role of coordination number and
curvature. The extent of the ferromagnetic phase in the dual-RBIM corresponds
to the correctable phase of hyperbolic surface codes under independent bit- and
phase-flip noise.
