In this article, for a rigid medium, we investigate the constitutive equations for the heat conduction given by a single and double phase-lag patterns, suggested by Tzou. Moreover, this research is extended to some Taylor expansions. Within this work, the thermodynamic compatibility of these systems is studied. In particular, we find that the models considered by Tzou are not in agreement with the laws of thermodynamics. We have observed that this compatibility can be proved for the first-order Taylor expansions of the SPL equation. In the following, we have investigated other models obtained by different order expansions of the phase lag patterns. Also, for these new differential systems we have studied the relationships between instability and thermodynamic incompatibility. Sufficient conditions for the onset of temperature instability are given. A uniqueness result for the ill-posed problem is obtained.