Most Comparator implementations do no such thing, but one scenario that might reasonably occur is using a SimpleDateFormat to compare Strings that represent dates. Unfortunately, SimpleDateFormat itself is not thread safe. Comparator is an interface, it has no inherent concurrency properties.
Is multithreaded thread-safe?
Thread safe: Implementation is guaranteed to be free of race conditions when accessed by multiple threads simultaneously. Conditionally safe: Different threads can access different objects simultaneously, and access to shared data is protected from race conditions.
Are collections thread-safe?
All collection classes (except Vector and Hashtable) in the java. util package are not thread-safe. The only two legacy collections are thread-safe: Vector and Hashtable.