  Understanding the thermodynamic properties of high-$T_c$ cuprate
superconductors is a key step to establish a satisfactory theory of these
materials. The electronic specific heat is highly unconventional, distinctly
non-BCS, with remarkable doping-dependent features extending well beyond $T_c$.
The pairon concept, bound holes in their local antiferromagnetic environment,
has successfully described the tunneling and photoemission spectra. In this
article, we show that the model explains the distinctive features of the
entropy and specific heat throughout the temperature-doping phase diagram.
Their interpretation connects unambiguously the pseudogap, existing up to
$T^*$, to the superconducting state below $T_c$. In the underdoped case, the
specific heat is dominated by pairon excitations, following Bose statistics,
while with increasing doping, both bosonic excitations and fermionic
quasiparticles coexist.
