Calculation of the density of a mixture for a given temperature, pressure, and composition from multi-parameter Helmholtz-energy-explicit mixture models (often referred to as the GERG formulation) sometimes fails; failures are caused by insufficiently accurate estimations of the density root, but also by insufficiently robust numerical methods that may not converge to the desired density solution. Furthermore, generally only one density solution is located at a time. Polynomial expansions have the characteristic that all roots of the expansion can be obtained reliably. Therefore the approach we propose is to develop a very good approximation of the equation of state based on Chebyshev orthogonal polynomial expansions, and solve for all the roots of the Chebyshev expansion - proxies for the roots of the equation of state. In this paper, we limit ourselves to the case where the temperature is known; the method is generalizable to other types of thermodynamic calculations. For mixtures, this Chebyshev proxy rootfinding results in a density calculation that is almost guaranteed to yield the right solution. These tools make multi-parameter equations of state nearly as reliable as cubic equations of state. The computational penalty from the use of Chebyshev expansions can be overcome through the exploitation of parallelism, though that is not further discussed here. This method is implemented in C++11; the code is provided in the supplemental material. (C) 2018 Published by Elsevier B.V.