In this article the quasi-Gaussian entropy (QGE) theory has been extended toward statistical-mechanical models that describe the temperature dependence of thermodynamic properties of fluids at fixed density over a very large temperature range, up to 15 times the critical temperature. The system's phase space is divided into multiple regions, each of which has a "potential" energy distribution that can be described by a simple model, e.g., a Gamma distribution. The overall "potential" energy distribution, which is directly related to the residual Helmholtz free energy of the system, then is a "mixture" of Gamma distributions, each with, for example, a different value of the "minimum" potential energy. Several such multistate models for the free energy were derived and tested on a series of small molecules at various densities: the hard core Yukawa fluid, the Lennard-Jones fluid, argon, methane, ammonia, water, and the extended simple point charge (SPC/E) water model. In almost all systems, a Gamma mixture of Gamma distributions provides a very accurate thermal model over a large temperature range starting, at the coexistence line and applicable from ideal gas to dense liquid, even in the vicinity of the critical point. The shape of the "potential" energy distribution and its density dependence reflects the underlying molecular interactions, which are discussed by comparing different systems.