In the previous study, two Inside Temperature (ITdQ=0 and ITIM) methods for estimating the temperature distributions in steady wave fronts in a thermoviscous material were established and the ITIM method was shown qualitatively to be effective for shock compressions where the effect of viscosity was distinguished. In this paper, these two methods are applied to the shock compressions of Yittria-doped Tetragonal Zirconia (YTZ) that is a thermoviscous material with a multiple shock Hugoniot. The YTZ Hugoniot consists of three partial curves including two kinks, that are the Hugoniot Elastic Limit (HEL) and the phase transition point. The shock temperatures evaluated by the ITIM method were close to the accurate temperatures obtained by the Walsh-Christian method in the whole stress range to 140 GPa examined here. Furthermore, the inside temperature distributions were approximately accurate because the effect of viscosity was distinguished in the shock compression. By these facts, it was considered that the fundamental assumption and the assumption on heat transport used in the ITIM method were valid and as a result, this method was effective. In addition, the influence of heat transport on the temperatures and thermoelastic stresses was examined.