You can use relationships between different tori. For instance, your torus lives in a long exact sequence

$T\to \mathbb G_m (\mathbb Q(i)) \to \mathbb G_m (\mathbb Q)$

This gives you a cohomology long exact sequence which relates your class number to the class numbers and unit groups of the fields. In this case, I think the class number is $2$, corresponding to the twist $x^2+ y^2=-1$.

You can probably do this in general but you need a spectral sequence.