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Recently, there has been an increase in efforts to understand how large language models (LLMs) propagate and amplify social biases. Several works have utilized templates for fairness evaluation, which allow researchers to quantify social biases in the absence of test sets with protected attribute labels. While template evaluation can be a convenient and helpful diagnostic tool to understand model deficiencies, it often uses a simplistic and limited set of templates. In this paper, we study whether bias measurements are sensitive to the choice of templates used for benchmarking. Specifically, we investigate the instability of bias measurements by manually modifying templates proposed in previous works in a semantically-preserving manner and measuring bias across these modifications. We find that bias values and resulting conclusions vary considerably across template modifications on four tasks, ranging from an 81% reduction (NLI) to a 162% increase (MLM) in (task-specific) bias measurements. Our results indicate that quantifying fairness in LLMs, as done in current practice, can be brittle and needs to be approached with more care and caution.