The phenomenon of the crossing points in specific-heat curves C(T,U) versus T of the one-dimensional half-filled-band asymmetric Hubbard model is investigated within the method of small-cluster exact-diagonalization calculations with extrapolation techniques to the infinite chain. A definite region where the crossing points occur is found to exist in a certain range of system parameters R (relative magnitude of the transfer integrals) and U (local interaction). We identify an association between crossing point occurrence and gradual localization of fermions.