  A microscopic thermodynamically consistent approach is applied to compute
electron capture (EC) rates and cross sections on nuclei in hot stellar
environments. The cross section calculations are based on the Donnelly-Walecka
multipole expansion method for treatment of semi-leptonic processes in nuclei.
To take into account thermal effects, we express the electron capture cross
section in terms of temperature- and momentum-dependent spectral functions for
respective multipole charge-changing operators. The spectral functions are
computed by employing the self-consistent thermal quasiparticle RPA (TQRPA)
with the Skyrme effective interaction. Three different Skyrme parametrizations
(SkM$^*$, SGII and SLy4) are used to investigate thermal effects on EC for
$^{56}$Fe and $^{78}$Ni. For $^{56}$Fe, the impact of thermally unblocked
GT$_+$ transitions on EC is discussed and the results are compared with those
from shell-model calculations. In particular, it is shown that for some
temperature and density regimes the TQRPA rates exceed the shell-model rates
due to violation of the Brink-Axel hypothesis within the TQRPA. For
neutron-rich $^{78}$Ni the full momentum-dependence of multipole transition
operators is considered and it is found that not only thermally unblocked
allowed $1^+$ transitions but also thermally unblocked first-forbidden $1^-$
and $2^-$ transitions favour EC.
