Simultaneous Localization and Mapping (SLAM) is a critical task that enables autonomous vehicles to construct maps and localize themselves in unknown environments. Recent breakthroughs combine SLAM with 3D Gaussian Splatting (3DGS) to achieve exceptional reconstruction fidelity. However, existing 3DGS-SLAM systems provide insufficient throughput due to the need for multiple training iterations per frame and the vast number of Gaussians. In this paper, we propose AGS, an algorithm-hardware co-design framework to boost the efficiency of 3DGS-SLAM based on the intuition that SLAM systems process frames in a streaming manner, where adjacent frames exhibit high similarity that can be utilized for acceleration. On the software level: 1) We propose a coarse-then-fine-grained pose tracking method with respect to the robot's movement. 2) We avoid redundant computations of Gaussians by sharing their contribution information across frames. On the hardware level, we propose a frame covisibility detection engine to extract intermediate data from the video CODEC. We also implement a pose tracking engine and a mapping engine with workload schedulers to efficiently deploy the AGS algorithm. Our evaluation shows that AGS achieves up to $17.12\times$, $6.71\times$, and $5.41\times$ speedups against the mobile and high-end GPUs, and a state-of-the-art 3DGS accelerator, GSCore.