Greenhouse gas (GHG) emission metrics, that is, conversion factors to evaluate the emissions of non-CO2 climate forcers on a common scale with CO2, serve crucial functions upon the implementation of the Paris Agreement. While different metrics have been proposed, they have not been investigated under a range of pathways, including those significantly overshooting the temperature targets of the Paris Agreement. Here we show that cost-effective metrics that minimize the overall cost of climate mitigation are time-dependent, primarily determined by the period remaining before the eventual stabilization, and strongly influenced by temperature overshoot. Our study suggests that flexibility should be maintained to adapt the choice of metrics in time as the future unfolds, if cost-effectiveness is a key consideration for global climate policy, instead of hardwiring the 100-year Global Warming Potential (GWP100) as a permanent feature of the Paris Agreement implementation as is currently under negotiation.