A Dyson equation approach is used to investigate chiral-symmetry breaking at finite temperatures for an effective QCD Hamiltonian. Coupled equations for the quasiparticle energy and chiral gap angle are derived. The resulting gap equation is no longer form invariant under shifts of the effective quark-quark potential by a constant. Furthermore, under such shifts of the potential, the quasiparticle energy no longer shifts by a constant. We show that this apparent shortcoming is connected with the fact that color fluctuations arise in performing calculations using the grand canonical ensemble without prior projection onto color-singlet states. We also investigate the identification of an effective quasiparticle mass at finite temperatures via the limit of the real-time single-particle Green's function at low momenta, in order to make clear the connection with the work of previous authors.