We present a theoretical study of the compressibility κ in a Fermi gas with attractive contact interactions, providing predictions for the strongly attractive regime and the superfluid phase. Our work emphasizes the compressibility sum rule and gauge invariance as constraints on κ, and we show how within a Gor'kov-based t-matrix approach, these can be satisfied in the normal phase when no approximations are made. For tractability, approximations must be introduced, and it is believed that thermodynamical approaches to κ are more reliable than correlation-function-based schemes. Contrasting with other studies in the literature, we present thermodynamic calculations of κ. As expected of an approximate approach, ours do not satisfy the compressibility sum rule. However, without adjusting parameters, they yield semiquantitative agreement with experiment and provide physical insight into similar results obtained via quantum Monte Carlo simulations. © 2013 American Physical Society.