Topology optimization is a computational technique that optimally allocates materials within a design space based on performance criteria, such as compliance or thermal resistance. However, traditional methods rely heavily on finite element analysis (FEA) for performance evaluation, which can be computationally expensive, especially for complex high-resolution models, limiting rapid prototyping and iterative design processes essential to advancing manufacturing and empowering enthusiasts and hobbyists. Recent advancements in operator learning offer an efficient alternative by directly learning mappings between infinite-dimensional spaces, e.g., differential operators, significantly reducing computational time. This research introduces the Convolutional Recurrent Operator Approximator Network (CRONet) as a surrogate model for FEA in topology optimization. CRONet efficiently approximates 2D and 3D operator mappings, enabling rapid, high-fidelity optimization with reduced computational costs. CRONet integrates both convolutional and recurrent neural architectures to capture spatial correlations and temporal dependencies within the optimization process. The recurrent component is crucial to enable CRONet learn from the sequential nature of topology optimization iterations, effectively capturing temporal relationships between successive design updates. This enables CRONet to adapt and maintain accurate predictions across iterations, further improving efficiency and robustness. Through benchmark case studies—the Messerschmitt-Bölkow-Blohm (MBB) beam and the cantilever beam—CRONet demonstrates up to a 78% reduction in computational time while retaining compliance accuracy within 5% of traditional FEA solutions. This novel approach promises a transformative impact on the design workflow, making topology optimization accessible for complex, manufacturable structures.