Residual specific heatscv/*are calculated from a Weeks-Chandler-Andersen-type perturbation theory for Lennard-Jones (LJ) and two-centre-Lennard-Jones (2CLJ) liquids. For spherical molecules the theory predicts that as the density is increased up to slightly above three times the critical density (ρc),cv/*increases to 2R. Corresponding states considerations indicate that rotational degrees of freedom in 2CLJ fluids make no contributions tocv/*. Comparing with real liquids, the theoretical predictions ofcv/*agree, within the accuracy of the experiments, with experimental data for liquid argon, methane, and oxygen, at densities ρ/ρc> 2·5. For ethane, the theoretical predictions ofcv/*, assuming a 2CLJ fluid with elongationL= 0·67, are too low. This deficiency in the theory for long molecules is attributed to the neglect of the angle-dependence of the background correlation function. However, computer simulation results forcv/*in 2CLJ fluids of elongationL= 0·67 agree with the perturbation theory predictions for the spherical LJ liquid at corresponding states. Again, for ρ/ρc> 2·5, both these latter results agree with the experimental values for ethane, within or nearly within the experimental accuracy.