  We study the energy spectrum and quantum Hall effects of the twisted double
bilayer graphene in uniform magnetic field. We investigate two different
arrangements, AB-AB and AB-BA, which differ in the relative orientation but
have very similar band structures in the absence of a magnetic field. For each
system, we calculate the energy spectrum and quantized Hall conductivities at
each spectral gap by using a continuum Hamiltonian that satisfies the
magneto-translation condition. We show that the Hofstadter butterfly spectra of
AB-AB and AB-BA stackings differ significantly, even though their zero magnetic
field band structures closely resemble; the spectrum of AB-AB has valley
degeneracy, which can be lifted by applying interlayer potential asymmetry,
while the spectrum of AB-BA has no such degeneracy in any case. We explain the
origin of the difference from the perspectives of lattice symmetry and band
topology.
