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 it's obvious (you always get the structure sheaf). This actually proves it's a line bundle in the etale topology, but that's the same as being a line bundle in the Zariski topology.
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 $T$.