Using a Frhlich Hamiltonian for the electron-lattice interaction, an expression for the Peierls phase transition temperature (Tp) of armchair carbon nanotube (ABT(n)) has been derived.As an illustration, the formula is used to estimate Tp of ABT(8) and ABT(10). The results indicate that Tp of ABT(n) will decrease with increasing of their diameters,and they are stable against the Peierls distortion and preserve metallic property at and far bellow room temperature.