  We investigated theoretically the specific heat of graphene, bilayer graphene
and twisted bilayer graphene taking into account the exact phonon dispersion
and density of states for each polarization branch. It is shown that contrary
to a conventional believe the dispersion of the out-of-plane acoustic phonons -
referred to as ZA phonons - deviates strongly from a parabolic law starting
from the frequencies as low as ~100 1/cm. This leads to the frequency-dependent
ZA phonon density of states and the breakdown of the linear dependence of the
specific heat on temperature T. We established that ZA phonons determine the
specific heat for T<200 K while contributions from both in-plane and
out-of-plane acoustic phonons are dominant for 200 K < T < 500 K. In the
high-temperature limit, T>1000 K, the optical and acoustic phonons contribute
approximately equally to the specific heat. The Debye temperature for graphene
and twisted bilayer graphene was calculated to be around ~1861 - 1864 K. Our
results suggest that the thermodynamic properties of materials such as bilayer
graphene can be controlled at the atomic scale by rotation of the sp2-carbon
planes.
