A new approach to the high temperature expansion of heat capacity C(T) is proposed. The lattice heat capacity is described by two first terms of this expansion with characteristic temperatures (theta(2) and theta(4)) plus a remainder. The latter is approximated by some function with only parameter theta* which is attributed to the upper limit of a phonon spectrum. This new form of expansion extends downwards along the range of a good approximation of a lattice heat capacity in comparison with ordinary truncated high temperature expansion. This, in turn, allows investigation of the new phenomena in some substances. This method was tested on several model curves C(T) with the phonon spectrum known in advance. The examples of the determination of phonon and electron characteristics for dielectrics, metals and high temperature superconductors are shown.