See Theorem 1.2 of the paper by [Bennett and Skinner] 1, which settles the problem for $p\ge 7$ (take there $C=1$ and $\alpha_0=2$).  (Earlier work of Darmon and Granville (using Faltings's theorem) showed that there are only finitely many solutions.)

See Theorem 1.2 of the paper by [Bennett and Skinner] 1, which settles the problem for $p\ge 7$ (take there $C=1$ and $\alpha_0=2$). Note that the Bennett-Skinner results are more general. (Earlier work of Darmon and Granville (using Faltings's theorem) showed that there are only finitely many solutions; again for more general such equations.)

Finally GH from MO has kindly pointed out an earlier paper of Darmon that handles this particular equation (assuming Shimura-Taniyama) for $p=11$ or $p\ge 17$.