  We have calculated electron capture rates for neutron-rich $N=50$ nuclei
($^{78}$Ni, $^{82}$Ge, $^{86}$Kr, $^{88}$Sr) within the Thermal QRPA approach
at temperatures $T=0$, corresponding to capture on the ground-state, and at
$T=10$ GK (0.86 MeV), which is a typical temperature at which the $N=50$ nuclei
are abundant during a supernova collapse. In agreement with recent experiments,
we find no Gamow-Teller (GT$_+$) strength at low excitation energies, $E<7$
MeV, caused by Pauli blocking induced by the $N=50$ shell gap. At the
astrophysically relevant temperatures this Pauli blocking of the GT$_+$
strength is overcome by thermal excitations across the $Z=40$ proton and $N=50$
neutron shell gaps, leading to a sizable GT contribution to the electron
capture. At the high densities, at which the $N=50$ nuclei are important for
stellar electron capture, forbidden transitions contribute noticeably to the
capture rate. Our results indicate that the neutron-rich $N=50$ nuclei do not
serve as an obstacle of electron capture during the supernova collapse.
