I checked on the Internet, it's all out there, sometimes it is "That hurt" and sometimes it's "That hurts", so which one is correct?
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"That hurts" is the present tense, and "That hurt" the past, so in theory there's a clear difference. In practice, of course, anything that hurts now (eg a slap in the face) did hurt when it happened; and if it happened recently enough for "That hurt" to be relevant it's probably still hurting. But it is better to decide which you meant, and then use the correct form: clear language goes with clear thought.
Both are correct. The difference is in the tense you are using. Hurt is past simple of... well, to hurt