The equilibrium structures and dynamics of a nanoscale system are regulated by a complex potential energy surface (PES). This is a key target of theoretical calculations but experimentally elusive. We report the measurement of a key PES parameter for a model nanosystem: size-selected Au nanoclusters, soft-landed on amorphous silicon nitride supports. We obtain the energy difference between the most abundant structural isomers of magic number Au561 clusters, the decahedron and face-centred-cubic (fcc) structures, from the equilibrium proportions of the isomers. These are measured by atomic-resolution scanning transmission electron microscopy, with an ultra-stable heating stage, as a function of temperature (125-500 °C). At lower temperatures (20-125 °C) the behaviour is kinetic, exhibiting down conversion of metastable decahedra into fcc structures; the higher state is repopulated at higher temperatures in equilibrium. We find the decahedron is 0.040 ± 0.020 eV higher in energy than the fcc isomer, providing a benchmark for the theoretical treatment of nanoparticles.