  We have calculated the plasmon modes in graphene double layer structures at
finite temperatures, taking into account the inhomogeneity of the dielectric
background of the system. The effective dielectric function is obtained from
the solution of the Poisson equation of three-layer dielectric medium with the
graphene sheets located at the interfaces, separating the different materials.
Due to the momentum dispersion of the effective dielectric function, the intra-
and inter-layer bare Coulomb interactions in the graphene double layer system
acquires an additional momentum dependence--an effect that is of the order of
the inter-layer interaction itself. We show that the energies of the in-phase
and out-of-phase plasmon modes are determined largely by different values of
the spatially dependent effective dielectric function. The effect of the
dielectric inhomogeneity increases with temperature and even at high
temperatures the energy shift induced by the dielectric inhomogeneity and
temperature itself remains larger than the broadening of the plasmon energy
dispersions due to the Landau damping. The obtained new features of the plasmon
dispersions can be observed in frictional drag measurements and in inelastic
light scattering and electron energy-loss spectroscopies.
