Why Java does not allow overriding equals(Object) in an Enum?

Anything but return this == other would be counter intuitive and violate the principle of least astonishment. Two enum constants are expected to be equal if and only if they are the same object and the ability to override this behavior would be error prone.

Same reasoning applies to hashCode(), clone(), compareTo(Object), name(), ordinal(), and getDeclaringClass().

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The JLS does not motivate the choice of making it final, but mentions equals in the context of enums here. Snippet:

The equals method in Enum is a final method that merely invokes super.equals on its argument and returns the result, thus performing an identity comparison.

There is already provides a strong intuitive notion of what it means for instances (values) of an enum to be equal. Allowing the overloading the equals method would lead to that notion being violated, leading to unexpected behavior, bugs and so on.