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(abridged) The flux-weighted gravity-luminosity relationship (FGLR) of bluesupergiant stars (BSG) links their absolute magnitude to the spectroscopicallydetermined flux-weighted gravity log g = Teff^4. BSG are the brightest stars inthe universe at visual light and the application of the FGLR has become apowerful tool to determine extragalactic distances. Observationally, the FGLRis a tight relationship with only small scatter. It is, therefore, ideal to beused as a constraint for stellar evolution models. The goal of this work is toinvestigate whether stellar evolution can reproduce the observed FGLR and todevelop an improved foundation of the FGLR as an extragalactic distanceindicator. We use different grids of stellar models for initial masses between9 and 40 Msun, for metallicities between Z = 0.002 and 0.014, with and withoutrotation, computed with various mass loss rates during the red supergiantphase. For each of these models we discuss the details of post-main sequenceevolution and construct theoretical FGLRs by means of population synthesismodels which we then compare with the observed FGLR. In general, the stellarevolution model FGLRs agree reasonably well with the observed one. There are,however, differences between the models, in particular with regard to the shapeand width (scatter) in the flux-weighted gravity-luminosity plane. The bestagreement is obtained with models which include the effects of rotation andassume that the large majority, if not all the observed BSG evolve towards thered supergiant phase and only a few are evolving back from this stage. Theeffects of metallicity on the shape and scatter of the FGLR are small. (...)

The flux-weighted gravity-luminosity relationship of blue supergiant stars as a constraint for stellar evolution