The generalization of the geometric phase to the realm of mixed states is known as the Uhlmann phase. Recently, applications of this concept to the field of topological insulators have been made and an experimental observation of a characteristic critical temperature at which the topological Uhlmann phase disappears has also been reported. Here we study the case of the Uhlmann phase of a paradigmatic system such as the spin-j particle in the presence of a slowly rotating magnetic field at finite temperature in an exact analytical form. We find that the Uhlmann phase is given by the argument of a complex-valued second-kind Chebyshev polynomial of order 2 j . Correspondingly, the Uhlmann phase displays 2 j singularities, occurring at the roots of such polynomials which define the critical temperatures at which the system undergoes topological order transitions. Appealing to the argument principle of complex analysis, each topological order is characterized by a winding number, which happens to be 2 j for the ground state and decreases by unity each time the increasing temperature passes through a critical value. We hope this study encourages experimental verification of this phenomenon of thermal control of topological properties, as has already been done for the spin-1 /2 particle.