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We demonstrate a spurious contribution to low-frequency critical current noise in Josephson junctions—normally attributed to charge trapping in the barrier—arising from temperature instabilities inherent in cryogenic systems. These temperature fluctuations modify the critical current via its temperature dependence. Cross-correlations between measured temperature and critical current noise in Al-AlOx-Al junctions show that, despite excellent temperature stability, temperature fluctuations induce observable critical current fluctuations. Particularly, because 1/f critical current noise has decreased with improved fabrication techniques in recent years, it is important to understand and eliminate this additional noise source.