Frieza pretended to work with Frost only so he could cheap shot Frost for an easy elimination. However, Frieza is very powerful in his golden form, much more than Frost. Why did he go through the trouble of tricking Frost if he could've just easily eliminated him the traditional way? Did Frost really pose such a threat to Frieza?
Frieza is playing to win, simple as that. And the prize here is both survival and the Super Dragonballs. And using Frost helps accomplish that goal, along with being very much in-character for Frieza.
First, Frieza loves torture and just screwing with people. As far as he is concerned, others are just tools for him to use. And once the tool has lost it's usefulness, it can be discarded. In DBZ, Frieza didn't need any of the races he enslaved to be able to conquer a planet. He was more than capable of doing that on his own. He did it for ease, simplicity and just being a sadist. He enjoys it.
Second, in order to survive, the other universes need to lose more fighters than U7. By allying, even temporarily with Frost, he effectively adds another fighter to their side. This just makes it easier to ensure U7 can win.
Additionally, a theme we've seen come up quite a few times is the notion of conserving stamina. Frieza has tremendous power, but using it drains stamina. Using Frost doesn't. So by getting Frost to do some of the dirty work lets Frieza conserve power for later when he needs it.
Also, having an out-of-team ally plays to the desire to get the Super Dragonballs. While I can't find specifics on who from the winning team gets to make the wish on the Super Dragonballs, presumably if there is only one person left, they would get the wish. Using Frost would allow Frieza to eliminate members of the U7 team without earning the wrath of Goku, Vegeta, etc, who could potentially eliminate him. (Think about what happened when Frieza killed Krillin on Namek.) And I am sure Frieza wants that wish. Badly. And this helps him to become the last man standing.
It all is simply a means to an end. And a means that Frieza particularly enjoys.
From a story telling perspective, this also serves as a reminder of who Goku has been forced to ally with. The stakes are high. And when even Goku recognizes the need for Frieza's help and tolerates his brutality, it further shows the direness of the situation. It also sets up a possibility of betrayal later, reminding the audience that Goku's only threats aren't just Jiren and losing the tournament. He also has to watch out to avoid being back stabbed. Imagine what could happen if Frieza were to win and get a wish on the Super Dragonballs. That's a terrifying thought.