The phrase "with a cold eye" can be replaced with several equivalent expressions in order to convey the same meaning in a slightly different way. For example, one might say "dispassionately," "unemotionally," or "objectively," which all suggest an analytical, unbiased approach to a situation. Other options include "detachedly," "indifferently," and "impersonally," all of which imply a lack of personal investment or concern. The phrase "with a cold eye" can also be swapped out for more metaphorical alternatives, such as "through a lens of skepticism" or "with a critical eye," which suggest a conscious effort to evaluate something with care and precision.