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.
