  We study the influence of thermal fluctuations in the phase diagram of a
recently introduced two-dimensional phase field crystal model with an external
pinning potential. The model provides a continuum description of pinned lattice
systems allowing for both elastic deformations and topological defects. We
introduce a non-conserved version of the model and determine the ground-state
phase diagram as a function of lattice mismatch and strength of the pinning
potential. Monte Carlo simulations are used to determine the phase diagram as a
function of temperature near commensurate phases. The results show a rich phase
diagram with commensurate, incommensurate and liquid-like phases with a
topology strongly dependent on the type of ordered structure. A finite-size
scaling analysis of the melting transition for the $c(2 \times 2)$ commensurate
phase shows that the thermal correlation length exponent $\nu$ and specific
heat behavior are consistent with the Ising universality class as expected from
analytical arguments.
