On stability of Friedmann-Lema\^itre-Robertson-Walker solutions in doubled geometries

Motivated by the models of geometry with discrete spaces as additional dimensions we investigate the stability of cosmological solutions in models with two metrics of the Friedmann-Lema^itre-Robertson-Walker type. We propose an effective gravity action that couples the two metrics in a similar manner as in the bimetric theory of gravity and analyse whether standard solutions with identical metrics are stable under small perturbations.