During effusive volcanic crises, the eruption and propagation of lava flows pose a significant hazard to nearby populations, homes, and infrastructure. Consequently, timely lava flow forecasts are a critical need for volcano observatory and emergency management operations. Previous lava flow modeling tools are typically either too slow to produce timely forecasts, or are fast, but lack critical aspects of lava physics or important forecasting outputs. In particular, the strong thermal stratification present in laminar, high-Prandtl number flows has generally been neglected. Bulk rheological changes have previously been computed from cell-averaged temperatures, assuming that the flow is thermally mixed. Here, we detail the development and initial testing of Lava2d, a new two-dimensional depth-averaged finite volume model of lava flow propagation over natural terrain which accounts for bulk rheological changes due to thermorheological stratification. We use a novel approach to energy conservation based on tracking cooling and solidifying at the flow base and at the moving flow surface, allowing for the estimation of more realistic vertical thermorheological profiles, while maintaining computational efficiency, producing very timely model runs. We validate our approach with three examples: comparison with theoretical propagation of crust-dominated lava flows, comparison with a large-scale molten basalt experiment from the Syracuse University Lava Project, and efficiency testing and comparison with the initial phase of the 1984 Mauna Loa lava flows. Our model is shown to produce rapid, realistic forecasts, making it a good candidate for operationalization in active volcanic regions such as in Hawai'i.