  It is conjectured that the average energy provides an upper bound on the rate
at which the complexity of a holographic boundary state grows. In this paper,
we perturb a holographic CFT by a relevant operator with a time-dependent
coupling, and study the complexity of the time-dependent state using the
\textit{complexity equals action} and the \textit{complexity equals volume}
conjectures. We find that the rate of complexification according to both of
these conjectures has UV divergences, whereas the instantaneous energy is UV
finite. This implies that neither the \textit{complexity equals action} nor
\textit{complexity equals volume} conjecture is consistent with the conjectured
bound on the rate of complexification.
