  We evaluate the full time dependence of holographic complexity in various
eternal black hole backgrounds using both the complexity=action (CA) and the
complexity=volume (CV) conjectures. We conclude using the CV conjecture that
the rate of change of complexity is a monotonically increasing function of
time, which saturates from below to a positive constant in the late time limit.
Using the CA conjecture for uncharged black holes, the holographic complexity
remains constant for an initial period, then briefly decreases but quickly
begins to increase. As observed previously, at late times, the rate of growth
of the complexity approaches a constant, which may be associated with Lloyd's
bound on the rate of computation. However, we find that this late time limit is
approached from above, thus violating the bound. Adding a charge to the eternal
black holes washes out the early time behaviour, i.e., complexity immediately
begins increasing with sufficient charge, but the late time behaviour is
essentially the same as in the neutral case. We also evaluate the complexity of
formation for charged black holes and find that it is divergent for extremal
black holes, implying that the states at finite chemical potential and zero
temperature are infinitely more complex than their finite temperature
counterparts.
