In this paper the HHCE is inspected on a microscopic level from a physical point of view. Starting from the Boltzmann transport equations we study the underlying approximations. We find that the hyperbolic approach to the heat current density violates the fundamental law of energy conservation. As a consequence, the HHCE predicts physically impossible solutions with a negative local heat content. This behaviour is demonstrated in detail for a standard problem in heat conduction, the solution for a point source.