Warm, high-CO₂ climates of Earth's past provide an opportunity to evaluate climate models under extreme forcing, and to explore mechanisms that lead to such warmth. One such time period is the early Eocene, when global mean surface temperatures were 10–17 °C higher than preindustrial, and CO₂ concentrations were ~1500 ppmv. In this paper we present the experimental design for Phase 2 of the Eocene component of the Deep-time Model Intercomparison project (DeepMIP-Eocene-p2). The aim is to provide a framework for modelling groups to carry out a common set of simulations, thereby facilitating exploration of inter-model dependencies. The focus is on the early Eocene Climatic Optimum (EECO, ~53.3–49.1 million years ago). Relative to Phase 1 of DeepMIP-Eocene, we provide a new paleogeography (topography, bathymetry) derived from several recent independent reconstructions that focused on different regions, a new vegetation derived by merging paleobotanical data with vegetation model simulations, and a new CO₂ specification derived from recent reevaluations of proxy data. The core set of simulations consists of a preindustrial control, an abrupt increase to 4x preindustrial CO₂ concentrations from this preindustrial control, a standard control EECO simulation at 5x preindustrial CO₂ concentrations, and an EECO simulation with preindustrial CO₂ concentrations. In addition to these core simulations, we suggest a suite of optional sensitivity studies, which allow the impact of various factors to be explored, such as topography/bathymetry, greenhouse gases, land-surface parameters, astronomical and solar forcings, and internal model parameters. The updated boundary conditions and guidance on initialisation and spinup in Phase 2 will allow more robust model-data comparisons, more accurate insights into mechanisms influencing early Eocene climate, and increased relevance for informing future climate change projections.