  (abridged) The flux-weighted gravity-luminosity relationship (FGLR) of blue
supergiant stars (BSG) links their absolute magnitude to the spectroscopically
determined flux-weighted gravity log g = Teff^4. BSG are the brightest stars in
the universe at visual light and the application of the FGLR has become a
powerful tool to determine extragalactic distances. Observationally, the FGLR
is a tight relationship with only small scatter. It is, therefore, ideal to be
used as a constraint for stellar evolution models. The goal of this work is to
investigate whether stellar evolution can reproduce the observed FGLR and to
develop an improved foundation of the FGLR as an extragalactic distance
indicator. We use different grids of stellar models for initial masses between
9 and 40 Msun, for metallicities between Z = 0.002 and 0.014, with and without
rotation, computed with various mass loss rates during the red supergiant
phase. For each of these models we discuss the details of post-main sequence
evolution and construct theoretical FGLRs by means of population synthesis
models which we then compare with the observed FGLR. In general, the stellar
evolution model FGLRs agree reasonably well with the observed one. There are,
however, differences between the models, in particular with regard to the shape
and width (scatter) in the flux-weighted gravity-luminosity plane. The best
agreement is obtained with models which include the effects of rotation and
assume that the large majority, if not all the observed BSG evolve towards the
red supergiant phase and only a few are evolving back from this stage. The
effects of metallicity on the shape and scatter of the FGLR are small. (...)
