We study the importance to the frequency of cooperation of the choice of updating strategies in a game played asynchronously or synchronously across layers in a multiplex network. Updating asynchronously in the public goods game leads to higher frequencies of cooperation compared to synchronous updates. How large this effect is depends on the sensitivity of the game dynamics to changes in the number of cooperators surrounding a player, with the largest effect observed when players payoffs are small. The discovery of this effect enhances understanding of cooperation on multiplex networks, and demonstrates a new way to maintain cooperation in these systems.