  We derive the low-temperature behavior of the Casimir-Polder free energy for
a polarizable atom interacting with graphene sheet which possesses the nonzero
energy gap $\Delta$ and chemical potential $\mu$. The response of graphene to
the electromagnetic field is described by means of the polarization tensor in
the framework of Dirac model on the basis of first principles of thermal
quantum field theory in the Matsubara formulation. It is shown that the thermal
correction to the Casimir-Polder energy consists of three contributions. The
first of them is determined by the Matsubara summation using the polarization
tensor defined at zero temperature, whereas the second and third contributions
are caused by an explicit temperature dependence of the polarization tensor and
originate from the zero-frequency Matsubara term and the sum of all Matsubara
terms with nonzero frequencies, respectively. The asymptotic behavior for each
of the three contributions at low temperature is found analytically for any
value of the energy gap and chemical potential. According to our results, the
Nernst heat theorem for the Casimir-Polder free energy and entropy is satisfied
for both $\Delta > 2\mu$ and $\Delta < 2\mu$. We also reveal an entropic
anomaly arising in the case $\Delta = 2\mu$. The obtained results are discussed
in connection with the long-standing fundamental problem in Casimir physics
regarding the proper description of the dielectric response of matter to the
electromagnetic field.
