If $X/T$ is actually a nice scheme, and $T$ acts freely, then you can think of this as follows: the pushforward of the structure sheaf on $X$ is etale locally isomorphic to the pushforward of the structure sheaf on $X/T \times T$, so proving that your sheaf is line bundle can be reduced to the case of a trivial bundle, in which case its obvious (you always get the structure sheaf).
This actually all works in the land of Artin stacks. In more abstract language, if you have any character of the torus, there's a line bundle on the stack $pt/T$ corresponding to this character (since a quasi-coherent sheaf on $pt/T$ is a vector bundle with $T$ action). The line bundles you're talking about are the pullback by the map $X/T\to pt/T$. This is essentially tautological; the stack $X/T$ is specifically designed so that a quasi-coherent sheaf on $X/T$ is a quasi-coherent sheaf on $X$ (of the same rank) which is $T$-equivariant. You're seeing that the structure sheaf can be $T$-equivariant in a bunch of different ways, since you can always tensor the action with a character of $T$.