The fate of a Fermi liquid on the verge of an instability towards formation of order is a long-standing problem in condensed matter physics. A canonical example of this phenomenon is an Ising nematic critical point, believed to be at play in several different correlated materials. The specific heat is a central thermodynamic property that can be used to characterize the properties of the quantum critical point. Here, we investigate the specific heat in a sign-free model that exhibits an Ising nematic quantum critical point, using quantum Monte Carlo simulations. At low temperatures near the quantum critical point, we observe a rapid rise of C/T, followed by a drop to zero as the system becomes superconducting. The onset of the rise in C/T starts at a temperature only 3-5 times larger than the superconducting T_c, suggesting that superconductivity and non-Fermi liquid behavior (as seen in an enhancement of the effective mass) onset at comparable energy scales. We support these conclusions with an analytical perturbative calculation within the RPA approximation.