Geometric frustration refers to the incommensurability between locally preferred order and global geometry. The inclusion of such frustration in systems of self-assembling particles has been shown to give rise to unique, scale-dependent states characterized by the self-limitation of domain size and the presence of topologically defective ground states. In this dissertation, we introduce a minimal lattice model of geometrically frustrated assembly and use a variety of numerical and theoretical techniques to explore its behavior at finite temperature. In chapter \ref{chapter: intro}, we review the literature of frustrated assembly and identify the key questions that we seek to address throughout this dissertation. In chapter \ref{chapter: model_intro}, we introduce our minimal model and develop the numerical and theoretical machinery that we will use throughout the following chapters. In addition to this, we will derive several key predictions for the effect of temperature on frustrated assembly. In chapter \ref{chapter: self-limiting assembly} and \ref{chapter: bulk condensation}, we use our numerical techniques to test these predictions and explore the self-limiting and defect bulk phase of assembly under fairly dilute conditions. In chapter \ref{chapter: equilibrium paths}, we investigate the role of entropy in stabilizing self-limiting assembly. After that, we relax the dilute restriction and study our model over the entire range of allowable concentration. Here, we show the existence of a percolation transition at high concentration and compare the structure of this resultant phase to the defective bulk structure. In chapter \ref{chapter: soft gauge model}, we generalize our model to allow the effects of subunit elasticity to be studied. Finally, in chapter \ref{chapter: conclusion}, we summarize the key results of this work and discuss several future directions that are motivated by experiment.