  Thermal properties of graphene monolayers are studied by path-integral
molecular dynamics (PIMD) simulations, which take into account the quantization
of vibrational modes in the crystalline membrane, and allow one to consider
anharmonic effects in these properties. This system was studied at temperatures
in the range from 12 to 2000~K and zero external stress, by describing the
interatomic interactions through the LCBOPII effective potential. We analyze
the internal energy and specific heat and compare the results derived from the
simulations with those yielded by a harmonic approximation for the vibrational
modes. This approximation turns out to be rather precise up to temperatures of
about 400~K. At higher temperatures, we observe an influence of the elastic
energy, due to the thermal expansion of the graphene sheet. Zero-point and
thermal effects on the in-plane and "real" surface of graphene are discussed.
The thermal expansion coefficient $\alpha$ of the real area is found to be
positive at all temperatures, in contrast to the expansion coefficient
$\alpha_p$ of the in-plane area, which is negative at low temperatures, and
becomes positive for $T \gtrsim$ 1000~K.
