  Consistent account is provided of strongly anisotropic properties in
thermodynamics of MgB$_2$ under magnetic field $H$ using approximate analytic
solution in the quasiclassical formalism. With the strength of the pairing
interactions and the Fermi velocities of two bands as fitting parameters, it is
possible to reproduce in wide regions of the phase diagram the $H$- and
temperature dependences of relevant quantities such as the zero-energy density
of states (ZEDOS), two energy gaps, the specific heat and the $H_{c2}$
anisotropy. It is demonstrated that the velocity ratio of two bands affects
considerably the low-field behavior of the ZEDOS and energy gaps. The ratio
also affects the high-temperature behavior of the $H_{c2}$ anisotropy. The
anisotropy of the coherence length in the $\sigma$ band depends weakly on $H$
and $T$, whereas the coherence length in the $\pi$ band remains isotropic even
in the presence of the inter-band coupling.
