The spotted lanternfly (SLF) is an invasive pest that has become established across much of the northeastern United States and threatens several significant crops, including apple, grape, and hop. As a cold-blooded organism, its population dynamical behavior is primarily dictated by climatic variables, with temperature being particularly influential. Predicting the behavior of this species in diverse climates is essential to the planning of effective interception and mitigation strategies. Here, we compute the SLF reproductive number (R_0) using a stage-age-structured system of PDEs representing the dynamics of a single population. To do this, we first develop a novel moving mesh method for computing solutions of the PDE system that captures age-advection accurately at coarse numerical discretizations. We then regard the dominant eigenvalue of a one-year numerical solution operator as a proxy for R_0. Using temperature-dependent rates of development, fecundity, and mortality fitted to emerging data, we compute R_0 as a function of the parameters that define seasonal temperature patterns. We discuss the establishment potential across the United States predicted by our parameter sweeps, address implications for control implementation, and preview future work.