Motivated by an example of Shih [11], we compute the fundamental gap of a family of convex domains in the hyperbolic plane H 2 , showing that there are convex domains for which, where D is the diameter of the domain and λ 1 , λ 2 are the first and second Dirichlet eigenvalues of the Laplace operator on the domain. The result contrasts with the case of domains in R n or S n , where , 10, 8, 5]. We also show that the fundamental gap of the domains in Shih's example is still greater than 3 2 π 2 D 2 , even though the first eigenfunction of the Laplace operator is not log-concave.