  We derive exact results for the rate of change of thermodynamic quantities,
in particular the configurational specific heat at constant volume, $C_V$,
along configurational adiabats (curves of constant excess entropy
$S_{\textrm{ex}}$). Such curves are designated isomorphs for so-called Roskilde
liquids, in view of the invariance of various structural and dynamical
quantities along them. Their slope in a double logarithmic representation of
the density-temperature phase diagram, $\gamma$ can be interpreted as one third
of an effective inverse power-law potential exponent. We show that in liquids
where $\gamma$ increases (decreases) with density, the contours of $C_V$ have
smaller (larger) slope than configurational adiabats. We clarify also the
connection between $\gamma$ and the pair potential. A fluctuation formula for
the slope of the $C_V$-contours is derived. The theoretical results are
supported with data from computer simulations of two systems, the Lennard-Jones
fluid and the Girifalco fluid. The sign of $d\gamma/d\rho$ is thus a third key
parameter in characterizing Roskilde liquids, after $\gamma$ and the
virial-potential energy correlation coefficient $R$. To go beyond isomorph
theory we compare invariance of a dynamical quantity, the self-diffusion
coefficient along adiabats and $C_V$-contours, finding it more invariant along
adiabats.
