  The Hofstadter-Hubbard model captures the physics of strongly correlated
electrons in an applied magnetic field, which is relevant to many recent
experiments on Moir\'e materials. Few large-scale, numerically exact
simulations exists for this model. In this work, we simulate the
Hubbard-Hofstadter model using the determinant quantum Monte Carlo (DQMC)
algorithm. We report the field and Hubbard interaction strength dependence of
charge compressibility, fermion sign, local moment, magnetic structure factor,
and specific heat. The gross structure of magnetic Bloch bands and band gaps
determined by the non-interacting Hofstadter spectrum is preserved in the
presence of $U$. Incompressible regions of the phase diagram have improved
fermion sign. At half filling and intermediate and larger couplings, a strong
orbital magnetic field delocalizes electrons and reduces the effect of Hubbard
$U$ on thermodynamic properties of the system.
