The extrapolation of small-cluster exact-diagonalization calculations is used to study temperature properties of the one-dimensional Falicov-Kimball model in the weak coupling limit. The results obtained are used to examine the specific heat (C) and the specific-heat coefficient (γ) as functions of temperature (τ=k B T) and f-level energy E f. The special attention is devoted to the analysis of the model in the metallic region (that exists only in the weak-coupling limit) in order to answer the question whether the Falicov-Kimball model can or can not exhibit the Fermi-liquid behavior in this region. Our results obtained from the analysis of the specific heat and the specific-heat coefficient indicate that the Falicov-Kimball model does not exhibit the Fermi liquid behaviour in the metallic phase. Importance of these results for the description of real materials (e.g., SmB6) is discussed.