  We investigate the effect of a constant magnetic field background on the
scalar QED pair production in a four-dimensional de Sitter spacetime ($\dsf$).
We have obtained the pair production rate which agrees with the known Schwinger
result in the limit of Minkowski spacetime and with the Hawking radiation in de
Sitter spacetime (dS) in the zero electric field limit. Our results describe
how the cosmic magnetic field affects the pair production rate in cosmological
setups. In addition, using the zeta function regularization scheme we have
calculated the induced current and examined the effect of a magnetic field on
the vacuum expectation value of the current operator. We find that, in the case
of a strong electromagnetic background the current responds as $E\cdot B$,
while in the infrared regime, it responds as $B/E$, which leads to a phenomenon
of infrared hyperconductivity. These results of the induced current have
important applications for the cosmic magnetic field evolution.
