  The temperature dependence of stellar electron-capture (EC) rates is
investigated, with a focus on nuclei around $N=50$, just above $Z=28$, which
play an important role during the collapse phase of core-collapse supernovae
(CCSN). Two new microscopic calculations of stellar EC rates are obtained from
a relativistic and a non-relativistic finite-temperature quasiparticle
random-phase approximation approaches, for a conventional grid of temperatures
and densities. In both approaches, EC rates due to Gamow-Teller transitions are
included. In the relativistic calculation contributions from first-forbidden
transitions are also included, and add strongly to the EC rates. The new EC
rates are compared with large-scale shell model calculations for the specific
case of $^{86}$Kr, providing insight into the finite-temperature effects on the
EC rates. At relevant thermodynamic conditions for core-collapse, the
discrepancies between the different calculations of this work are within about
one order of magnitude. Numerical simulations of CCSN are performed with the
spherically-symmetric GR1D simulation code to quantify the impact of such
differences on the dynamics of the collapse. These simulations also include EC
rates based on two parametrized approximations. A comparison of the neutrino
luminosities and enclosed mass at core bounce shows that differences between
simulations with different sets of EC rates are relatively small ($\approx
5\%$), suggesting that the EC rates used as inputs for these simulations have
become well constrained.
