This work investigates the stabilized inelastic response of structures subjected to cyclic thermomechanical loading when elastic properties depend on temperature. Classical Melan-Koiter shakedown theory is commonly formulated assuming constant elastic coefficients; however, for industrial components experiencing large thermal cycles this assumption may become questionable. A two-bar elastic–perfectly plastic system is analyzed, where the Young’s modulus of one of the bars varies with temperature, while in the second bar Young’s modulus remains constant. The asymptotic regimes of elastic response, shakedown, reverse (alternating) plasticity, and plastic ratcheting are identified and represented in a Bree-type diagram as functions of mechanical force and thermal amplitude. Overall, the study highlights the additional complexity introduced by temperature-dependent elastic coefficients and provides a consistent analytical–numerical framework for predicting stabilized cyclic responses. A classical step-by-step incremental cyclic analysis predicts the transient evolution over multiple cycles and classifies the stabilized response by monitoring the cycle-to-cycle evolution of plastic strain. Shakedown is detected when the plastic strain increment per cycle vanishes, reverse plasticity when a stable closed hysteresis loop develops without net drift and ratcheting when progressive plastic strain accumulation occurs. A key outcome is that, when elasticity depends on temperature, residual stress and strain fields may remain time-dependent even in shakedown. However, when only the asymptotic stabilized response is sought, a direct method, which is computationally much faster than any incremental step by step one, may be employed. Such an approach is the Residual Stress Decomposition Method (RSDM). In this procedure, stresses are decomposed into thermoelastic, and residual components and stabilized fields are represented using a truncated Fourier series. The method has been used successfully in cases of constant elastic properties exhibiting stable and rapid convergence. In this research work, the method is shown that it may be used to cater for structures, having temperature-dependent elastic properties.