If there was a time or an issue that Republicans could stand for and actually be pro-market as they keep saying they are (in theory), it's this one. They'd probably get as much support as they usually get from evangelists, which seems to be their core audience these days, or at least that's how they're acting in public.
Make no mistake, the reason they would be in favor of this at all is that copyright holders (Hollywood) are perceived to favor Democrats. It's another bullet.
It's nice that this issue happens to put them on the "right" side.
The evangelicals who support the GOP tend to be poor, rural, and of low education. They'll support pretty much any fiscal policies the GOP takes as long as the party adheres to conservative social policies.