The climate is a nonequilibrium system undergoing the continuous action of forcing and dissipation. Under the effect of a spatially inhomogeneous absorption of solar energy, all the climate components dynamically respond until an approximate steady state (or attractor) is reached. However, multiple steady states can coexist for a given forcing and with the same boundary conditions. Here, we apply the Thermodynamic Diagnostic Tool (TheDiaTo) to investigate the statistical properties of five coexisting climates, ranging from a snowball to an ice-free aquaplanet, obtained in MITgcm coupled simulations. The aim is to explore the multistability of the climate model setup by highlighting differences in competing steady states and their characteristic signatures regarding the meridional transport of heat and water mass, the Lorenz energy cycle, and the material entropy production. We also investigate how such attractors change when the model configuration is varied. We consider, in particular, the effect of changing the representation of the cloud albedo, and of implementing an improved closure of the energy budget. We find that, even if the dynamics remain on the same attractor, state variables are modified. The set of metrics in TheDiaTo quantifies such modifications and represents a valuable tool for model evaluation.