  We find analytic asymptotic expressions at low temperature for the Casimir
free energy, entropy and pressure of two parallel graphene sheets in the
framework of the Lifshitz theory. The reflection coefficients of
electromagnetic waves on graphene are described on the basis of first
principles of quantum electrodynamics at nonzero temperature using the
polarization tensor in (2+1)-dimensional space-time. The leading contributions
to the Casimir entropy and to the thermal corrections to the Casimir energy and
pressure are given by the thermal correction to the polarization tensor at
nonzero Matsubara frequencies. It is shown that the Casimir entropy for two
graphene sheets goes to zero when the temperature vanishes, i.e., the third law
of thermodynamics (the Nernst heat theorem) is satisfied. At low temperature,
the magnitude of the thermal correction to the Casimir pressure between two
graphene sheets is shown to vary inversely proportional to the separation. The
Nernst heat theorem for graphene is discussed in the context of problems
occurring in Casimir physics for both metallic and dielectric plates.
