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We use density functional perturbation theory based on the pseudo-potential to calculate the phonon spectrum, phonon density of states, specific heat capacity and mechanical properties of pristine and cobalt doped (5,0) single wall carbon nanotube (CNT). In the calculations, we consider one Co atom in the center of the unit cell of the tube and it is shown that the pristine (5,0) CNT is nonmagnetic while the Co-doped tube becomes magnetic. Young's modulus for both systems is about 1 TPa (after Co-doping it goes slightly higher) and the Poisson ratio for the pristine tube becomes quite a bit larger than the doped one. On the other hand, the calculated value of radial breath mode for the pristine CNT is in good agreement with the experimental reports while after Co-doping it is increased. In addition, heat capacity of the doped CNT is reduced, which leads to some important empirical applications.