Are there many hyperbolic structures on $ I \times$ Torus?

Question

I have a basic question on Thurston's hyperbolization in the (probably too simple) case of $M=[0,1] \times T$ where $T$ is a torus. According to Thurston on page 359, M has no hyperbolic structure of finite volume. However

1. The metric $ds^2=dt^2 + e^{2t} |dz^2|$ has negative curvature ($R=-6$) and finite volume. What am I not thinking right here?

2. Is there a unique infinite-volume hyperbolic structure on M? If not, is there a parameterization of these hyperbolic structures?

I have also seen in Marden's book Hyperbolic Manifolds the same statement as that of Thurston.

Any comment is welcome. Thanks.