  We perform a three-dimensional, finite temperature Skyrme-Hartree-Fock study
of inhomogeneous nuclear matter to determine the critical density and
temperature for the phase transition between the pasta phase and homogeneous
matter and its properties. We employ four different parametrizations of the
Skyrme nuclear energy-density functional, SkM$^*$, SLy4, NRAPR and SQMC700,
which span a range of saturation-density symmetry energy behaviours constrained
by a variety of nuclear experimental probes. For each of these interactions we
calculate free energy, pressure, entropy and chemical potentials in the range
of particle number densities where the nuclear pasta phases are expected to
exist, 0.02 - 0.12 fm$^{-3}$, temperatures 2 - 8 MeV and a proton fraction of
0.3. We find unambiguous evidence for a first-order phase transition to uniform
matter, unsoftened by the presence of the pasta phases. No conclusive signs of
first-order phase transition between the pasta phases is observed, and it is
argued that the thermodynamic quantities vary continuously right up to the
first-order phase transition to uniform matter. We compare our results with
thermodynamic spinodals calculated using the same Skyrme parameterizations,
finding that the effect of short-range Coulomb correlations and quantum shell
effects included in our model leads to the pasta phases existing at densities
up to 0.1 fm$^{-3}$ above the spinodal boundaries, thus increasing the
transition density to uniform matter by the same amount. The transition density
is otherwise shown to be insensitive to the symmetry energy at saturation
density within the range constrained by the concordance of a variety of
experimental constraints, and can be taken to be a well-determined quantity.
