The dielectric breakdown properties of hot SF6 gas during the dielectric recovery phase play an important role in understanding gas breakdown occurring in high-voltage circuit breakers. This paper is devoted to the theoretical investigation of dielectric breakdown properties of hot SF6 gas contaminated by copper at temperatures of 300-3500 K and pressures of 0.01-1.6 MPa. The equilibrium compositions of SF6-Cu mixtures are obtained with the consideration of condensed species. The unknown ionization cross sections for CuS, CuF and CuF2 are calculated using a Deutsch-Märk (DM) formalism based on quantum chemistry. The two-term Boltzmann equation is adopted to numerically calculate the electron energy distribution function, collision ionization coefficient and electron attachment coefficient. Then the reduced critical electric field strength is determined when the effective ionization coefficient equals to zero. The influences of the Cu proportion and gas pressure on the dielectric breakdown properties are investigated. It is shown that the existence of copper compounds increases the concentration of high-energy electrons significantly, even for the case with a very low percentage (e.g. 1% Cu). With the increase of copper content, the value of (E/N)cr is reduced remarkably at temperatures below 3000 K, but enhanced slightly above 3000 K. It is also found that the increase of pressure can improve the dielectric breakdown performance of hot SF6-Cu mixtures.