  The phonon modes at finite temperature in graphene and single-walled carbon
nanotubes (SWCNT) are investigated by the mode projection technique combined
with molecular dynamics simulation. It is found that the quadratic phonon
spectrum of the flexural mode in graphene is renormalized into linear at finite
temperature by the phonon-phonon scattering. Possible influence of this
renormalization on the electron-phonon interaction and phonon thermal transport
is analyzed. For the SWCNT, however the quadratic property of the phonon
spectrum for the flexural mode can survive at finite temperature under
protection of the two-fold degeneracy of the flexural mode in SWCNT due to its
quasi-one-dimensional structure. The frequency and life time of optical phonon
modes are also studied at different temperatures. In particular, the life time
of the in-plane optical phonons in graphene and the axial optical phonon in
SWCNT can be described by an uniform formula $\tau (T)=560/T$, which coincides
with the experimental results.
