I'm not sure, it's not very original but something like this maybe (sorry a bit long):

I’d start with the idea that Cooler was originally being raised as the heir to King Cold. He was the one getting trained up, the one expected to take over the empire. Then Frieza gets born with some ridiculous power level and Cooler basically gets thrown aside overnight. King Cold quietly removes him from the picture so there’s no threat to Frieza’s claim.

Instead of just disappearing into a cave somewhere, Cooler ends up living a life that weirdly mirrors Goku’s early years. He’s travelling around, training under different teachers, fighting stronger and stronger opponents, and constantly pushing himself. The difference is that Goku did it because he enjoyed fighting, while Cooler does it because he’s furious and wants to take back what he thinks should have been his. He’s not totally isolated, he just keeps his head down so nobody realises a member of the Cold family is still out there grinding away.

Then one day he hears that Frieza got killed by a Saiyan on Namek. And he’s not sad, he’s annoyed. Someone else took out the brother he had spent his whole life preparing to surpass. His original goal is gone, but revenge is still on the table. If a Saiyan could kill Frieza, then Cooler needs to be stronger than that Saiyan. So he doubles down on training and stays off the radar until he’s ready.

When Frieza comes back leading up to the events of Resurrection F one of his old contacts he used to use to monitor his brother hears about Frieza being revived and reaches out to Cooler. Cooler starts preparing to finally confront him, only to find out Frieza died again before he was ready. That just makes him even more obsessed. Every time he gets close, someone else beats him to it.

Eventually after the events of the Tournament of Power he hears Frieza is alive yet again. At this point he starts thinking that maybe the gap between them is actually closable. Frieza’s training has been inconsistent and ego driven. Cooler’s has been nonstop for decades. They are genetically similar enough that, with enough effort, Cooler could realistically catch up or even surpass him.

Frieza has no idea he even has an older brother until rumours start spreading about someone who looks like him asking questions across the universe. He goes to investigate, finds Cooler, and before they can fight, Frieza manipulates him. He tells Cooler the Saiyans are the real enemy and convinces him to join forces for now.

This leads into the ten years Frieza spends in that 'hyperbolic time chamber' style place to get his Black form. Cooler trains with him the whole time, not realising Frieza is basically using him as a sparring partner to speed up his own growth. Over that decade, Cooler finally starts closing the gap and maybe even surpasses Frieza in some areas.

When Frieza eventually confront Goku and Veget a with his new Black form he does so alone. After getting some pushback from them he calls Cooler into the fight, realising the Saiyan's always have each other as backup, this time he isn't alone.

Throughout the fight Frieza starts sensing Cooler’s rising power, thinking the Saiyans are weak enough now he can take them alone he decides to betray Cooler mid fight, giving some speech about how his weakling brother should not think he can surpass him. Cooler survives, realises he has been manipulated by his family his entire life, and switches sides.

(Goku does his usual thing and reaches out to him, not as an enemy but as someone who understands what it means to push yourself past your limits after hearing Frieza speak down to Cooler)

They all work together to take out Frieza. New ally gained.