Motivated by finding models that admit a solution to the current Hubble constant tension, we implement a general extension to Lambda-CDM. We first review the needed machinery to model the constituents of the universe to linear order in perturbation theory before introducing generalized dark matter. We then present our numerical implementation of high-dimensional generalized dark matter (HDGDM) with a redshift-dependent equation of state parameter. Following statistical analysis of the model with CMB, BAO, and supernova data, we demonstrate that marginalized posterior distributions can easily lead to misleading conclusions on the viability of a high-dimensional model such as this one. We highlight problems with standard techniques and corresponding mitigation strategies. The subsequent results find an observationally favorable parameter space region, making near-future testable predictions. We finally present a technique of dimension reduction for high-dimensional models that focuses on changing statistical predictions of a target parameter.