  We consider the moduli space of log smooth pairs formed by a cubic surface
and an anticanonical divisor. We describe all compactifications of this moduli
space which are constructed using Geometric Invariant Theory and the
anticanonical polarization. The construction depends on a weight on the
divisor. For smaller weights the stable pairs consist of mildly singular
surfaces and very singular divisors. Conversely, a larger weight allows more
singular surfaces, but it restricts the singularities on the divisor. The
one-dimensional space of stability conditions decomposes in a wall-chamber
structure. We describe all the walls and relate their value to the worst
singularities appearing in the compactification locus. Furthermore, we give a
complete characterization of stable and polystable pairs in terms of their
singularities for each of the compactifications considered.
