  We report the excitation energy dependence of specific heat (\cv) of hadronic
matter at freeze-out in Au+Au and Cu+Cu collisions at the Relativistic Heavy
Ion Collider energies by analyzing the published data on event-by-event mean
transverse momentum (\meanpt) distributions. The \meanpt~distributions in
finite \pt~ranges are converted to distributions of effective temperatures, and
dynamical fluctuations in temperature are extracted by subtracting widths of
the corresponding mixed event distributions. The heat capacity per particle at
the kinetic freeze-out surface is presented as a function of collision energy,
which shows a sharp rise in \cv~below \sNN~=~62.4~GeV. We employ the Hadron
Resonance Gas (HRG) model to estimate \cv~at the chemical and kinetic
freeze-out surfaces. The experimental results are compared to the HRG and other
theoretical model calculations. HRG results show good agreement with data.
Model predictions for \cv~at the Large Hadron Collider energy are presented.
