  The electron capture process plays an important role in the evolution of the
core collapse of a massive star that precedes the supernova explosion. In this
study, the electron capture on nuclei in stellar environment is described in
the relativistic energy density functional framework, including both the finite
temperature and nuclear pairing effects. Relevant nuclear transitions $J^\pi =
0^\pm, 1^\pm, 2^\pm$ are calculated using the finite temperature proton-neutron
quasiparticle random phase approximation with the density-dependent
meson-exchange effective interaction DD-ME2. The pairing and temperature
effects are investigated in the Gamow-Teller transition strength as well as the
electron capture cross sections and rates for ${}^{44}$Ti and ${}^{56}$Fe in
stellar environment. It is found that the pairing correlations establish an
additional unblocking mechanism similar to the finite temperature effects, that
can allow otherwise blocked single-particle transitions. Inclusion of pairing
correlations at finite temperature can significantly alter the electron capture
cross sections, even up to a factor of two for ${}^{44}$Ti, while for the same
nucleus electron capture rates can increase by more than one order of
magnitude. We conclude that for the complete description of electron capture on
nuclei both pairing and temperature effects must be taken into account.
