  We present thermodynamic relationships between the free energy of the
phase-field crystal (PFC) model and thermodynamic state variables for bulk
phases under hydrostatic pressure. This relationship is derived based on the
thermodynamic formalism for crystalline solids of Larch\'e and Cahn [Larch\'e
and Cahn, Acta Metallurgica, Vol. 21, 1051 (1973)]. We apply the relationship
to examine the thermodynamic processes associated with varying the input
parameters of the PFC model: temperature, lattice spacing, and the average
value of the PFC order parameter, $\bar{n}$. The equilibrium conditions between
bulk crystalline solid and liquid phases are imposed on the thermodynamic
relationships for the PFC model to obtain a procedure for determining
solid-liquid phase coexistence. The resulting procedure is found to be in
agreement with the method commonly used in the PFC community, justifying the
use of the common-tangent construction to determine solid-liquid phase
coexistence in the PFC model. Finally, we apply the procedure to an
eighth-order-fit (EOF) PFC model that has been parameterized to
body-centered-cubic ($bcc$) Fe [Jaatinen et al., Physical Review E 80, 031602
(2009)] to demonstrate the procedure as well as to develop physical intuition
about the PFC input parameters. We demonstrate that the EOF-PFC model
parameterization does not predict stable $bcc$ structures with positive vacancy
densities. This result suggests an alternative parameterization of the PFC
model, which requires the primary peak position of the two-body direct
correlation function to shift as a function of $\bar{n}$.
