The correlations of the free-energy landscape of mean-field spin glasses at different temperatures are investigated, concentrating on models with a first-order freezing transition. Using a ‘potential function’ we follow the metastable states of the model in temperature, and discuss the possibility of level crossing (which we do not find) and multifurcation (which we find). The dynamics at a given temperature starting from an equilibrium configuration at a different temperature is also discussed. In the presence of multifurcation, we find that the equilibrium is never achieved, leading to an aging behaviour at slower energy levels than usual aging. The relevance of the observed mechanisms for real structural glasses is discussed, and some numerical simulations of a soft sphere model of glass are presented.