  Twisted two-dimensional structures open new possibilities in band structure
engineering. At magic twist angles, flat bands emerge, which give a new drive
to the field of strongly correlated physics. In twisted double bilayer graphene
dual gating allows changing the Fermi level and hence the electron density and
also allows tuning the interlayer potential, giving further control over band
gaps. Here, we demonstrate that by applying hydrostatic pressure, an additional
control of the band structure becomes possible due to the change of tunnel
couplings between the layers. We find that the flat bands and the gaps
separating them can be drastically changed by pressures up to 2 GPa, in good
agreement with our theoretical simulations. Furthermore, our measurements
suggest that in finite magnetic field due to pressure a topologically
non-trivial band gap opens at the charge neutrality point at zero displacement
field.
