Primate cortical evolution has been characterized by massive and disproportionate expansion of a set of specific regions in the neocortex. The associated increase in neocortical neurons comes with a high metabolic cost, thus the functions served by these regions must have conferred significant evolutionary advantage. Here, across a series of experiments, we show that the evolutionary high-expanding hotspots - as estimated from patterns of evolutionary expansion from several primate species - share functional connections with different brain networks in a context-dependent manner. This capacity of the hotspots to connect flexibly with various specialized brain networks depending on particular cognitive requirements suggests that their selective growth and sustainment in evolution has been linked to their involvement in supramodal cognition. In accordance with an evolutionary-developmental view, we find that this ability to flexibly modulate functional connections as a function of cognitive state emerges gradually through childhood, with a prolonged developmental trajectory plateauing in young adulthood.