  We note that the standard inverse system volume scaling for finite-size
corrections at a first-order phase transition (i.e., 1/L^3 for an L x L x L
lattice in 3D) is transmuted to 1/L^2 scaling if there is an exponential
low-temperature phase degeneracy. The gonihedric Ising model which has a
four-spin interaction, plaquette Hamiltonian provides an exemplar of just such
a system. We use multicanonical simulations of this model to generate
high-precision data which provides strong confirmation of the non-standard
finite-size scaling law. The dual to the gonihedric model, which is an
anisotropically coupled Ashkin-Teller model, has a similar degeneracy and also
displays the non-standard scaling.
