By laminating piezoelectric and flexible materials during the manufacturing process, we can improve the performance of electronic devices. In smart structure design, it is also important to understand how the functionally graded piezoelectric (FGP) structure changes over time when thermoelasticity is assumed. This is because these structures are often exposed to both moving and still heat sources during many manufacturing processes. Therefore, it is necessary to conduct theoretical and experimental studies of the electrical and mechanical characteristics of multilayer piezoelectric materials when they are subjected to electromechanical loads and heat sources. Since the infinite speed of heat wave propagation is a challenge that classical thermoelasticity cannot address, other models based on extended thermoelasticity have been introduced. For this reason, the effects of an axial heat supply on the thermomechanical behavior of an FGP rod using a modified Lord-Shulman model with the concept of a memory-dependent derivative (MDD) will be explored in this study. The exponential change of physical properties in the direction of the axis of the flexible rod will be taken into account. It was also assumed that there is no electric potential between the two ends of the rod while it is fixed at both ends and thermally isolated. Applying the Laplace transform method, the distributions of the physical fields under investigation were calculated. The obtained results were compared to those in the corresponding literature with varying heterogeneity values, kernel functions, delay times, and heat supply speeds. It was discovered that the studied physical fields and the dynamic behavior of the electric potential are weakened by increasing the inhomogeneity index.