The study of nonlinear problems associated with the process of heat transfer in material is very important for practice. Previously, the authors proposed an effective algorithm for determining the volumetric heat capacity and thermal conductivity of a substance based on the results of experimental observations of the dynamics of the temperature field in an object or the dynamics of heat flows at the boundary of the domain. In this work, a new cost functional is used to determine temperature-dependent thermodynamic parameters, which is a weighted sum of the deviation of temperature in the domain and heat fluxes at the boundary of the domain from the experimental values. The use of the new functional showed a noticeable increase in the efficiency of the proposed algorithm and the reliability of the results obtained. The consideration is carried out on the basis of the first boundary value problem for a one-dimensional non-stationary heat equation. The inverse coefficient problem under consideration is reduced to a variational problem, which is solved by gradient methods based on the application of the Fast Automatic Differentiation technique. The question of the uniqueness of the solution to the inverse problem is investigated.