The voter model is a classical interacting particle system explaining consensus formation on a social network. Real social networks feature not only a heterogeneous degree distribution but also connections changing over time. We study the voter model on a rank one scale-free network evolving in time by each vertex \emph{updating} (refreshing its edge neighbourhood) at any rate $κ=κ(N)$. We find the dynamic giant component phase transition in the consensus time of the voter model: when $κ\ll \tfrac{1}{N}$, the subcritical graph parameters are slower by a factor of $\tfrac{N}{\log N}$. Conversely, when $κ\gg 1$ the effect of the giant is removed completely and so for either graph parameter case we see consensus time on the same order as in the static supercritical case (up to polylogarithmic corrections). The intermediate dynamic speeds produce consensus time for subcritical network parameters longer not by the previous factor $\tfrac{N}{\log N}$, but by the factor $\tfrac{1}κ$.