Rob Gronkowski Honors Avicii with 'Levels' Dance Battle

Everybody mourns in their own way ... for Rob Gronkowski, he paid his respects to Avicii by dancing his face off to the DJ's biggest hit -- and TMZ Sports has the video.

The New England Patriots tight end was at the Summit Rooftop Lounge in Austin, Texas over the weekend when Avicii's "Levels" started booming out of the speakers.

So, what did Gronk do? Dance battle.

The NFL star cleared an area and went toe-to-toe with a female clubgoer ... and together, they put on a show!