We consider the global cosmological evolution and the evolution of the density contrast in the Bose–Einstein condensate dark matter model, in the framework of a post-Newtonian cosmological approach. In the Bose–Einstein model, dark matter can be described as a non-relativistic, Newtonian gravitational condensate, whose density and pressure are related by a barotropic equation of state. For a condensate with quartic non-linearity, the equation of state is polytropic with index n= 1.The basic equation describing the evolution of the perturbations of the Bose–Einstein condensate is obtained, and its solution is studied by using both analytical and numerical methods. The global cosmological evolution as well as the evolution of the perturbations of the condensate dark matter shows significant differences with respect to the pressureless dark matter model, considered in the framework of standard cosmology. Therefore, the presence of condensate dark matter could have modified drastically the cosmological evolution of the early universe as well as the large-scale structure formation process.