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We study the impacts of thermal expansion, arising from the asymmetric interparticle potential, on thermal conductance in the FPU-α-β model. A nonmonotonic dependence of the temperature gradient and thermal conductance on the cubic interaction parameter α are shown, which corresponds to the variation of the coefficient of thermal expansion. Three domains with respect to α can be identified. The results are explained based on the detailed analysis of the asymmetry of the interparticle potential. The self-consistent phonon theory, which can capture the effect of thermal expansion, is developed to support our explanation in a quantitative way. Our result would be helpful to understand the issue that whether there exist normal thermal conduction in the FPU-α-β model.