  We implement the GW space-time method at finite temperatures, in which the
Green's function G and the screened Coulomb interaction W are represented in
the real space on a suitable mesh and in imaginary time in terms of Chebyshev
polynomials, paying particular attention to controlling systematic errors of
the representation. Having validated the technique by the canonical application
to silicon and germanium, we apply it to calculation of band gaps in hexagonal
solid hydrogen with the bare Green's function obtained from density functional
approximation and the interaction screened within the random phase
approximation (RPA). The band gap results, obtained from the asymptotic decay
of the full Green's function without resorting to analytic continuation,
suggest that the solid hydrogen above 270 GPa can not adopt the
hexagonal-closed-pack (hcp) structure. The demonstrated ability of the method
to store the full G and W functions in memory with sufficient accuracy is
crucial for its subsequent extensions to include higher orders of the
diagrammatic series by means of diagrammatic Monte Carlo algorithms.
