Optimization plays a central role in modern radiation therapy, where it is used to determine optimal treatment machine parameters in order to deliver precise doses adapted to each patient case. In general, solving the optimization problems that arise can present a computational bottleneck in the treatment planning process, as they can be large in terms of both variables and constraints. As high precision is often sought, second-order optimization algorithms, such as sequential quadratic programming (SQP) and/or interior point methods (IPM) are commonly used. Existing implementations of these algorithms often use direct linear solvers internally, and are typically intended to run on CPUs. Utilizing iterative linear solvers instead is an active research topic in the optimization community, and one which carries the potential to enable efficient GPU acceleration for these types of optimization problems. Numerical stability issues make this a difficult problem for optimization solvers targeting problems from a wide range of application areas, however. In this paper, we develop and implement a GPU-accelerated interior point method for optimization problems from radiation therapy using iterative linear algebra. We utilize a so called doubly augmented formulation of the Karush-Kuhn-Tucker linear systems, together with a Jacobi-preconditioned conjugate gradient solver, which is able to find sufficiently accurate search directions while running on GPU. By evaluating our solver on real optimization problems from a commercial treatment planning system for radiation therapy, we show that our method can accelerate the aggregated time-to-solution by 1.4 and 4.4 times, respectively, for two patient cases.