  We present a finite-temperature extension of the retarded cumulant Green's
function for calculations of exited-state and thermodynamic properties of
electronic systems. The method incorporates a cumulant to leading order in the
screened Coulomb interaction $W$ and improves excited state properties compared
to the $GW$ approximation of many-body perturbation theory. Results for the
homogeneous electron gas are presented for a wide range of densities and
temperatures, from cool to warm dense matter regime, which reveal several
hitherto unexpected properties. For example, correlation effects remain strong
at high $T$ while the exchange-correlation energy becomes small. In addition,
the spectral function broadens and damping increases with temperature, blurring
the usual quasi-particle picture. Similarly Compton scattering exhibits
substantial many-body corrections that persist at normal densities and
intermediate $T$. Results for exchange-correlation energies and potentials are
in good agreement with existing theories and finite-temperature DFT
functionals.
