A well-known feature of first-order phase transitions is that fixed boundary conditions can strongly influence finite-size corrections, modifying the leading corrections for an L3 lattice in 3d from order 1/L3 under periodic boundary conditions to 1/L. A rather similar effect, albeit of completely different origin, occurs when the system possesses an exponential low-temperature phase degeneracy of the form 23L which causes for periodic boundary conditions a leading correction of order 1/L2 in 3d. We discuss a 3d plaquette Hamiltonian (“gonihedric”) Ising model, which displays such a degeneracy and manifests the modified scaling behaviour. We also investigate an apparent discrepancy between the fixed and periodic boundary condition latent heats for the model when extrapolating to the thermodynamic limit.