  We show that for any liquid or solid with strong correlation between its
$NVT$ virial and potential-energy equilibrium fluctuations, the temperature is
a product of a function of excess entropy per particle and a function of
density, $T=f(s)h(\rho)$. This implies that 1) the system's isomorphs (curves
in the phase diagram of invariant structure and dynamics) are described by
$h(\rho)/T={\rm Const.}$, 2) the density-scaling exponent is a function of
density only, 3) a Gr{\"u}neisen-type equation of state applies for the
configurational degrees of freedom. For strongly correlating atomic systems one
has $h(\rho)=\sum_nC_n\rho^{n/3}$ in which the only non-zero terms are those
appearing in the pair potential expanded as $v(r)=\sum_n v_n r^{-n}$. Molecular
dynamics simulations of Lennard-Jones type systems confirm the theory.
