The log-likelihood function is deterministic. The gradient of the log-likelihood can change a lot when the gradient is evaluated at different points.
I guess the acceptance rate depend on the error of the integrator. If that is true, then using a step-size that adapts to local geometry would indirectly adapt the step-size to the acceptance rate.
I remember the RMHMC paper says it needs the inverse and gradient of the fisher information matrix. That seems slow. tfp says they can do a form of RMHMC.
I was thinking about http://www.unige.ch/~hairer/preprints/revstep.pdf