A temperature wave method has been applied to observe the thermal diffusivity through the isotropic (Iso)-nematic (N)-smectic Sm-A-Sm-C-Sm-B-crystals VI-VII-VIII phase transitions of terephthal-bis-(4-n-butylaniline) (TBBA). Critical anomalies have been found in the N-Sm-A and Sm-C-Sm-B phase transitions as diplike shaped, consistent with the predictions based on the dissipative couplings between the order parameter and the conserved free-energy density. Singular points with a gap have been observed at the Sm-B-crystal VI, crystals VI-VII, and crystals VII-VIII phase transitions, which show polymorphic behaviors on heating and cooling. The second-order Sm-A-Sm-C phase transition emerged as a singular temperature dependence. In all the phases thermal diffusivity decreases with increasing temperature except for Sm-C, where thermal diffusivity increases with increasing temperature. The origin of the anomaly in the thermal diffusivity in Sm-C is discussed based on the parametric analysis of dynamic critical behavior in the Sm-A-Sm-C phase transitions together with the tilt angle change obtained by use of simultaneous measurements of x-ray diffraction and differential scanning calorimetry.