  In this paper we study generalized time-fractional diffusion equations on the
Poincar\`e half plane $\mathbb{H}_2^+$. The time-fractional operators here
considered are fractional derivatives of a function with respect to another
function, that can be obtained by starting from the classical
Caputo-derivatives essentially by means of a deterministic change of variable.
We obtain an explicit representation of the fundamental solution of the
generalized-diffusion equation on $\mathbb{H}_2^+$ and provide a probabilistic
interpretation related to the time-changed hyperbolic Brownian motion. We
finally include an explicit result regarding the non-linear case admitting a
separating variable solution.
