The presence of burstiness in temporal social networks, revealed by a power-law form of the waiting time distribution of consecutive interactions, is expected to produce aging effects in the corresponding time-integrated network. Here, we propose an analytically tractable model, in which interactions among the agents are ruled by a renewal process, that is able to reproduce this aging behavior. We develop an analytic solution for the topological properties of the integrated network produced by the model, finding that the time translation invariance of the degree distribution is broken. We validate our predictions against numerical simulations, and we check for the presence of aging effects in a empirical temporal network, ruled by bursty social interactions.