Palaeoclimate reconstructions of periods with warm climates and high atmospheric CO2 concentrations are crucial for developing better projections of future climate change. Deep-ocean1,2 and high-latitude3 palaeotemperature proxies demonstrate that the Eocene epoch (56 to 34 million years ago) encompasses the warmest interval of the past 66 million years, followed by cooling towards the eventual establishment of ice caps on Antarctica. Eocene polar warmth is well established, so the main obstacle in quantifying the evolution of key climate parameters, such as global average temperature change and its polar amplification, is the lack of continuous high-quality tropical temperature reconstructions. Here we present a continuous Eocene equatorial sea surface temperature record, based on biomarker palaeothermometry applied on Atlantic Ocean sediments. We combine this record with the sparse existing data4-6 to construct a 26-million-year multi-proxy, multi-site stack of Eocene tropical climate evolution. We find that tropical and deep-ocean temperatures changed in parallel, under the influence of both long-term climate trends and short-lived events. This is consistent with the hypothesis that greenhouse gas forcing7,8, rather than changes in ocean circulation9,10, was the main driver of Eocene climate. Moreover, we observe a strong linear relationship between tropical and deep-ocean temperatures, which implies a constant polar amplification factor throughout the generally ice-free Eocene. Quantitative comparison with fully coupled climate model simulations indicates that global average temperatures were about 29, 26, 23 and 19 degrees Celsius in the early, early middle, late middle and late Eocene, respectively, compared to the preindustrial temperature of 14.4 degrees Celsius. Finally, combining proxy- and model-based temperature estimates with available CO2 reconstructions8 yields estimates of an Eocene Earth system sensitivity of 0.9 to 2.3 kelvin per watt per square metre at 68 per cent probability, consistent with the high end of previous estimates11.