  We report the first study on the thermal behaviour of the stiffness of
individual carbon nanotubes, which is achieved by measuring the resonance
frequency of their fundamental mechanical bending modes. We observe a reduction
of the Young's modulus over a large temperature range with a slope $-(173\pm
65)$ ppm/K in its relative shift. These findings are reproduced by two
different theoretical models based on the thermal dynamics of the lattice.
These results reveal how the measured fundamental bending modes depend on the
phonons in the nanotube via the Young's modulus. An alternative description
based on the coupling between the measured mechanical modes and the phonon
thermal bath in the Akhiezer limit is discussed.
