We have studied the finite-temperature domain-growth kinetics in two-dimensional models with high ground-state degeneracy and non-conserved order parameter by means of Monte Carlo temperature-quenching techniques. The models are Q-state Potts models with anisotropic pair potential which at low temperatures are found to develop vortex structures during the non-equilibrium domain-growth process. The vortex formation is an entropy-driven effect in this model. We have identified vortex-antivortex pairs as a significant structural element of the temporal domain-boundary morphology and shown that the vortices and anti-vortices annihilate during the growth process without pinning the ordering process. The resulting growth law as derived from the excess domain-wall energy and the vortex density is found to be the Lifshitz-Allen-Cahn law with the classical exponent value, n = ½, independent of the ordering degeneracy Q.