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One of the equilibrium temperature profiles along the conductor is found to be the unstable solution between well-known upper and lower solutions of the energy balance equation of high temperature superconducting (HTS) current leads. The transient behavior of the temperature profile of the HTS leads is analyzed by perturbing the unstable equilibrium temperature profile. The stability of the leads against thermal disturbances is explained by the use of results of the transient analysis. The stability of ideal gas cooled leads is compared with that of the conduction cooled leads.