In his brief, non-canon history in Dragon Ball Z, Cooler has always somehow been one step ahead of and one step behind his brother Frieza. Each of his movie appearances portrayed him as a more powerful opponent than Frieza. And just like his brother, Cooler has powered up via transformations, lost to Goku, failed to conquer Namek, returned as a cyborg, then lost again.
He can fail at anything Frieza fails at, but he does it cooler. It does, however, raise the question: If Cooler can do everything Frieza can do, does that mean he could achieve a Golden Form as well?
Aside from being canon, the only thing Frieza has done so far that Cooler hasn't is achieve the Golden Form. Though the evil siblings are said to be unusually powerful for their race, Frieza has always been hailed as the true prodigy -- which he demonstrated by only needing a couple of months to catch up with decades of training for Goku and Vegeta. This may just be from lack of opportunity, considering Cooler can't exactly train in Hell. It's unknown how powerful Cooler could become if he were able to train.
After the success of reimagining and canonizing Broly, Dragon Ball has confirmed that fans are open to the return of iconic villains from DBZ's past. However, bringing back Cooler just to resume his following in Frieza's footsteps would be a waste of his potential. It's a problem that's plagued Vegeta's and Goku's relationship until only recently in the Dragon Ball Super manga. So a more appropriate question might be, rather than could Cooler achieve a Golden form, should he?
The line between fan service and character development can be difficult to toe. Dragon Ball has so many fans that it's difficult to say what fan service even is anymore in this franchise. Part of the problem with Cooler is his long history of doing things Frieza has already done. It would be challenging to give him a Golden Form without it feeling like diving back into old habits that have made him a bit of a punchline. Seeing Cooler in a Golden Form would certainly be visually compelling, but it would also highlight, in gold, everything that is wrong with the character.
However, the question of whether or not Cooler is capable of the Golden Form is difficult to answer. If going by whether it's canon, then no, he is not, because canonically he does not even exist. The argument could be made that he is canon thanks to a cameo in Dragon Ball GT, but whether or not GT itself is canon has never been confirmed. The vast expanse of Dragon Ball non-canon material rivals that of the actual canon, making hypotheticals such as this pretty perplexing.
Since Cooler is a strictly non-canon character, the question posed requires the non-canon answer, which is yes. During the Prison Planet Saga of Super Dragon Ball Heroes, Cooler displays the ability to transform into a golden state like Frieza's. In a hilarious lack of history and awareness that almost feels like satire, he proudly declares that if Frieza can do it, so can he.