  We discuss a two-band model for two-dimensional superconductors with electron
and hole bands separated by an energy gap and singlet $d$-wave pairing in each
band. This model exhibits a V-shaped to U-shaped transition in the density of
the states of the superconductor, which is similar to that found in recent
tunneling experiments in magic-angle twisted trilayer graphene [H. Kim et al.,
Nature 606, 494 (2022)]. This qualitative difference in behavior occurs when
the electron and hole chemical potentials change, leading to topological
quantum phase transitions between gapless and gapful $d$-wave superconducting
states, due to the annihilation of chiral Dirac fermions at the critical
points. Lastly, we show that direct thermodynamic evidence of these topological
quantum phase transitions can be found in measurements of the compressibility,
which exhibits logarithmic singularities at the transition points.
