  We study Hofstadter bilayers, i.e. coupled hopping models on two-dimensional
square lattices in a perpendicular magnetic field. Upon tracing out one of the
layers, we find an explicit expression for the resulting entanglement spectrum
in terms of the energy eigenvalues of the underlying monolayer system. For
strongly coupled layers the entanglement Hamiltonian is proportional to the
energetic Hamiltonian of the monolayer system. The proportionality factor,
however, cannot be interpreted as the inverse thermodynamic temperature, but
represents a phenomenological temperature scale. We derive an explicit relation
between both temperature scales which is in close analogy to a standard result
of classic thermodynamics. In the limit of vanishing temperature, thermodynamic
quantities such as entropy and inner energy approach their ground-state values,
but show a fractal structure as a function of magnetic flux.
