  We present a model of pressure effects of a two-band superconductor based on
a Ginzburg-Landau free energy with two order parameters. The parameters of the
theory are pressure as well as temperature dependent. New pressure effects
emerge as a result of the competition between the two bands. The theory then is
applied to MgB2. We identify two possible scenaria regarding the fate of the
two $\sigma$ subbands under pressure, depending on whether or not both subbands
are above the Fermi energy at ambient pressure. The splitting of the two
subbands is probably caused by the E2g distortion. If only one subband is above
the Fermi energy at ambient pressure (scenario I), application of pressure
diminishes the splitting and it is possible that the lower subband participates
in the superconductivity. The corresponding crossover pressure and Gruneisen
parameter are estimated. In the second scenario both bands start above the
Fermi energy and they move below it, either by pressure or via the substitution
of Mg by Al. In both scenaria, the possibility of electronical topological
transition is emphasized. Experimental signatures of both scenaria are
presented and existing experiments are discussed in the light of the different
physical pictures.
