  We show that rhombohedral graphite may support surface superconductivity with
an unusual relation between the BCS coupling constant and the order parameter.
This feature results from the properties of the states localized on the
graphite surfaces. In a description including only the nearest neighbour
coupling of the graphene layers, the surface states are topologically protected
and have a flat band dispersion. We show that including higher order couplings
destroys this flat band character and leads to a particle-hole symmetry
breaking quadratic dispersion with a large effective mass. Employing this
dispersion, we then show its effect on superconductivity and find two regimes
of parameters, depending on the relation between the strength of the coupling
constant and the details of the quadratic dispersion. For low coupling
strengths, superconductivity is localized on the surfaces, but the order
parameter is exponentially suppressed as in a conventional BCS superconductor,
whereas for large coupling strengths we obtain surface superconductivity with a
linear relation between the order parameter and the coupling constant. Our
results may explain the recent findings of graphite superconductivity with a
relatively high transition temperature.
