The 'transient' response of the heat transfer coefficient is investigated in turbulent and relaminarising, unsteady flow. We find that the response to sinusoidal perturbations collapse on a single parameter, of, and we use this scaling to develop an empirical model for predicting the response to non-sinusoidal changes. The model works well under fully turbulent conditions but fails when the flow relaminarises. We find no evidence of any mechanism that would lead to heat transfer enhancement over and beyond a quasi-steady effect. On the contrary, relaminarisation appears to decrease the average heat transfer in some unsteady flows.