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Given the upcoming High-Luminosity LHC Upgrade, the performance requirements for the trigger systems associated with the LHC experiments will increase due to the larger volume of data to be processed. One of the possibilities that the ATLAS Collaboration is evaluating for upgrading the software-based portion of its trigger system is the use of Graphical Processing Units as hardware accelerators. The present work focuses on the GPU acceleration of the Topological Clustering algorithm, which is used to reconstruct calorimeter showers by grouping cells according to their signal-to-noise ratio. A more GPU parallelizable version of the Topological Clustering, called Topo-Automaton Clustering, was implemented within AthenaMT, the software framework of the ATLAS trigger, and its results were compared to those of the standard CPU algorithm to ensure physical validity is maintained. Time measurements suggest an average improvement of the event processing time by a factor between 3.5 and 5.5 (depending on the kind of the event), though less than 20% of that time corresponds to the algorithm itself, suggesting that the main bottleneck lies in data transfers and conversions.