The microwave cavity technique is currently the most sensitive way of looking for dark matter axions in the 0.1 GHz-10 GHz range, corresponding to masses of 0.41 µeV-41 µeV. A particular challenge for frequencies greater than 5 GHz is designing a cavity with a large volume that contains a resonant mode that shows high coupling to dark matter axions, a high quality factor, is broadly tunable, and is free from intruder modes. For the Haloscope at Yale Sensitive to Axion Cold dark matter, we have designed and constructed an optimized high frequency cavity with a tuning mechanism that preserves a high degree of rotational symmetry, critical to maximizing its figure of merit. This cavity covers an important frequency range according to recent theoretical estimates for the axion mass, 5.5 GHz-7.4 GHz, and the design appears extendable to higher frequencies as well. This paper will discuss key design and construction details of the cavity, present a summary of the design evolution, and alert practitioners of potentially unfruitful avenues for future work.