  In this paper, we use the "complexity equals action" (CA) conjecture to
evaluate the holographic complexity in some multiple-horzion black holes for
F(Riemann) gravity coupled to a first-order source-free electrodynamics.
Motivated by the vanishing result of the purely magnetic black hole founded by
Goto $et.\, al$, we investigate the complexity in a static charged black hole
with source-free electrodynamics and find that this vanishing feature of the
late-time rate is universal for a purely static magnetic black hole. However,
this result shows some unexpected features of the late-time growth rate. We
show how the inclusion of a boundary term for the first-order electromagnetic
field to the total action can make the holographic complexity be well-defined
and obtain a general expression of the late-time complexity growth rate with
these boundary terms. We apply our late-time result to some explicit cases and
show how to choose the proportional constant of these additional boundary terms
to make the complexity be well-defined in the zero-charge limit. For the static
magnetic black hole in Einstein gravity coupled to a first-order
electrodynamics, we find that there is a general relationship between the
proper proportional constant and the Lagrangian function $h(\math{F})$ of the
electromagnetic field: if $h(\math{F})$ is a convergent function, the choice of
the proportional constant is independent on explicit expressions of
$h(\math{F})$ and it should be chosen as $4/3$; if $h(\math{F})$ is a divergent
function, the proportional constant is dependent on the asymptotic index of the
Lagrangian function.
