  Specific heat of phonon cavities is investigated in order to analyse the
effect of phonon confinement on thermodynamic properties. The specific heat of
free standing very thin SiN membranes in the low dimensional limit is measured
down to very low temperatures (from 6~K to 50~mK). In the whole temperature
range, we measured an excess of specific heat orders of magnitude bigger than
the typical value observed in amorphous solids. Below 1~K, a cross-over in
$c_p$ to a lower power law is seen, and the value of specific heat of thinner
membranes becomes larger than that of thicker ones demonstrating a significant
contribution coming from the surface. We show that this high value of the
specific heat cannot be explained by the sole contribution of 2D phonon modes
(Lamb waves). The excess specific heat, being thickness dependent, could come
from tunneling two level systems (TLS) that form in low density regions of
amorphous solids located on the surfaces. We also show that the specific heat
is strongly tuned by the internal stress of the membrane by orders of
magnitude, giving unprecedentedly high values, making low stress SiN very
efficient for energy storage at very low temperature.
