I don't know exactly what the genre of this painting is. It looks nice to me, so I hope I could see more of it from other artist as well.
Formally it reminds me of art brut (or the related primitivism), because of its somewhat crude and naive style. It could also be considered expressionist, but it's hard to tell without seeing more art by the artist (and expressionism is a very broad art movement, which might not help you very much).
The texture and colour scheme are slightly reminiscent of Paul Klee's work, but the artwork you post is more naive and straightforward:
Paul Klee, Abendliche figur, 1935. Source.
The image's content is quite obviously symbolist, though: through easily legible symbols and representations we can deduct the artist's (possible) intention.
The main figure has symbols for a sun and a moon behind their left and right hand, respectively, a heart symbol at around the physical location of their heart, and their belly seemingly occupied by the head of another, squatting figure. Green leaves on both sides suggest a presence of and possible synergy with nature.
Since the shirt and the information I found about it online doesn't show an artist, I think this artwork was made by Rob Thomas of Matchbox Twenty himself, as the pieces depicted on other shirts in the Hard Rock Café Signature shirt series seem to have been created by their respective artists, as well.
Art brut or outsider art consists of works of art that are usually created by non-art-educated artists, who lack theoretical, historical, and/or formalistic education.