Hamiltonians with explicit temperature and density dependence are employed in classical and quantized partition functions to derive caloric and thermal equations of state (EoS) for real gases and liquids. To define a fluid equilibrium system, the density- and temperature-dependent Hamiltonian has to satisfy an equilibrium condition derived here, which ensures that the internal-energy derivative of entropy coincides with the reciprocal temperature. The inverse problem to obtain an effective Hamiltonian from a prescribed thermal EoS is discussed, and explicit examples of this reconstruction are given for cubic and non-cubic EoSs. A non-cubic multi-parameter EoS is proposed, a generalization of the Peng-Robinson EoS, which can accurately reproduce the power-law ascent of pressure isotherms in the high-pressure/high-density regime. This EoS is put to test by fitting isothermal data sets of classical and quantum fluids (nitrogen, carbon monoxide, methane, carbon dioxide, methanol, water, hydrogen and helium) over an extended pressure range up to 1 GPa, at critical, super- and subcritical temperatures. The least-squares fits are compared with the cubic Peng-Robinson approximation of the pressure isotherms, which becomes singular at high density. The internal-energy variable derived from the effective Hamiltonian can also be used to discriminate between different EoSs in the high-density regime, as demonstrated with methane isotherms. (C) 2019 Elsevier B.V. All rights reserved.