  Thermodynamic properties of the multiband superconductor MgB$_{2}$ have often
been described using a simple sum of the standard BCS expressions corresponding
to $\sigma$- and $\pi$-bands. Although, it is \textit{a priori} not clear if
this approach is working always adequately, in particular in cases of strong
interband scattering. Here we compare the often used approach of a sum of two
independent bands using BCS-like $\alpha$-model expressions for the specific
heat, entropy and free energy to the solution of the full Eliashberg equations.
The superconducting energy gaps, the free energy, the entropy and the heat
capacity for varying interband scattering rates are calculated within the
framework of two-band Eliashberg theory. We obtain good agreement between the
phenomenological two-band $\alpha$-model with the Eliashberg results, which
delivers for the first time the theoretical verification to use the
$\alpha$-model as a useful tool for a reliable analysis of heat capacity data.
For the thermodynamic potential and the entropy we demonstrate that only the
sum over the contributions of the two bands has physical meaning.
