  The $GW$ approximation is a well-established method for calculating
ionization potentials and electron affinities in solids and molecules. For
numerous years, obtaining self-consistent $GW$ total energies in solids has
been a challenging objective that is not accomplished yet. However, it was
shown recently that the linearized $GW$ density matrix permits a reliable
prediction of the self-consistent $GW$ total energy for molecules [F. Bruneval
et. al. J. Chem. Theory Comput. 17, 2126 (2021)] for which self-consistent $GW$
energies are available. Here we implement, test, and benchmark the linearized
$GW$ density matrix for several solids. We focus on the total energy, lattice
constant, and bulk modulus obtained from the $GW$ density matrix and compare
our findings to more traditional results obtained within the random phase
approximation (RPA). We conclude on the improved stability of the total energy
obtained from the linearized $GW$ density matrix with respect to the mean-field
starting point. We bring compelling clues that the RPA and the $GW$ density
matrix total energies are certainly close to the self-consistent $GW$ total
energy in solids if we use hybrid functionals with enriched exchange as a
starting point.
