Phase field modeling of fracture has become a popular framework for simulating crack propagation due to its ability to represent complex crack paths without explicit crack tracking. However, the presence of an inherent length scale parameter in the phase field formulation requires a uniformly fine spatial discretization, which significantly increases computational costs. The computational costs are exceptionally high in dynamic fracture simulations, where the global time step is dictated by the smallest element of the mesh. To address this problem, an asynchronous variational integrator (AVI) [1] has recently been proposed for a phase field model of dynamic fracture. The AVI framework allows individual spatial elements to evolve using independent characteristic time steps. In this approach, the mechanical fields are updated at every time step, while the phase field variable is updated less frequently, specifically following the displacement update of the largest spatial mesh element [2]. Building upon this formulation, spatial adaptivity is introduced by dynamically refining the mesh based on a criterion guided by the evolution of the phase field variable. The resulting framework achieves both spatial and temporal adaptivity: mesh refinement is driven by the phase field variable, while time step adaptivity is inherently provided by the AVI scheme. Numerical benchmark studies demonstrate that this spatio-temporally adaptive approach significantly reduces computational costs through local mesh refinement, selective elemental updates, and less frequent phase field updates, while exhibiting good energy behavior and high accuracy in capturing complex dynamic fracture phenomena. Keywords — brittle fracture, dynamic fracture, phase field model, asynchronous variational integrators, adaptivity, computational efficiency REFERENCES [1] Lew A., Marsden J.E., Ortiz M. & West M. Asynchronous Variational Integrators. Archive forRational Mechanics and Analysis 2003;167: 85–146. [2] Jadhav D. B., Phansalkar D., Weinberg K., Ortiz M., Leyendecker S. A New Approach to Asynchronous Variational Integrators for a Phase Field Model of Dynamic Fracture. International Journal for Numerical Methods in Engineering 2025; 126(6) : e70025.