  We show that one can obtain logarithmic improvements of $L^2$ geodesic
restriction estimates for eigenfunctions on 3-dimensional compact Riemannian
manifolds with constant negative curvature. We obtain a
$(\log\lambda)^{-\frac12}$ gain for the $L^2$-restriction bounds, which
improves the corresponding bounds of Burq, G\'erard and Tzvetkov, Hu, Chen and
Sogge. We achieve this by adapting the approaches developed by Chen and Sogge,
Blair and Sogge, Xi and the author. We derive an explicit formula for the wave
kernel on 3D hyperbolic space, which improves the kernel estimates from the
Hadamard parametrix in Chen and Sogge. We prove detailed oscillatory integral
estimates with fold singularities by Phong and Stein and use the Poincar\'e
half-space model to establish bounds for various derivatives of the distance
function restricted to geodesic segments on the universal cover $\mathbb{H}^3$.
