<p>This paper investigates the thermoelastic frictional contact behavior of functionally graded piezoelectric material (FGPM) coating structures with arbitrarily varying material properties, subjected to a rigid conductive cylindrical punch. Considering the effect of the thermal convection term during heat conduction, a homogeneous multi-layered model is employed to approximate the arbitrarily varying properties of FGPM coatings, which overcomes the mathematical limitations of traditional continuous models restricted to idealized exponential gradients. Based on thermoelastic theory, Fourier integral transform technique and transfer matrix method, the thermos-electric-elastic coupled contact problem is converted into a system of coupled Cauchy singular integral equations of the first and second kinds. The least squares method and an iterative technique are employed for numerical solutions. The effects of the friction coefficient and gradient index on the thermal frictional contact characteristics are discussed in detail. Notably, the inclusion of the thermal convection term rigorously captures the directional heat advection, resulting in an asymmetric temperature field shifted towards the trailing edge. Additionally, the regulatory role of gradient types of coating material properties on the surface tensile stress and temperature is analyzed. The results indicate a clear performance compromise: power-law gradient distribution is optimal for reducing surface tensile stresses, while sinusoidal gradient offers superior thermal insulation effect. These findings indicate that appropriate adjustment of the coating gradient index, friction coefficient, and gradient types can effectively alleviate the thermo-electro-elastic contact damage on the material surface, which provides a theoretical reference for the damage-resistant design of piezoelectric materials.</p>