This three-manifold can also be constructed by taking a genus two surface $S$, crossing with the interval $I$ to get $S \times I$, and attaching a pair of one-handles both of which connect $S \times \{0\}$ to $S \times \{1\}$. Applying the Klein combination theorem gives infinitely many complete hyperbolic metrics on the result.
To see this: Note that the boundary (blue) is compressible along two curves (the compressing disks meet the orange and the red, respectively). After compressing, the remains of the blue is a pair of genus two surfaces. Now find enough annuli (some will need to cross the green, or purple, or both) to obtain the product.