Some oozes are ageless or live a very long time

Notably, Formless spawns:

Formless spawn are ageless and sexless, all of them having spawned from sources deep below the ground. The original formless spawn issued fully formed and aware from Tsathoggua himself, but after a time, the rocks near where he slumbered seemed to become capable of creating them. When a formless spawn masquerades as a simple pool, its metabolism slows significantly and it can stay for centuries without eating, but upon awakening, it is always ravenous and risks starving to death if it cannot find food

Shoggoths, for certain:

Shoggoths are said to have been around for eons. Legends also claim that, when compared to any other oozes, shoggoths have much greater wisdom. Shoggoths were once the servants of dark powers, but cast off their masters after centuries of mistreatment.[1] Sages believe that aboleths might be their creators, but this is incorrect: the elder things, an even more ancient race, were behind their existence, and uprisings of shoggoth slaves who develop intelligence have been the doom for countless elder thing colonies

Colors Out of Space are so alien that we cannot understand if a single creature dies to spawn another, or if they simply duplicate themselves to reproduce, like several other oozes.

Over the course of several weeks, months, or even years, the colour feeds upon the surrounding plant and animal life—the act of being fed upon is weirdly addictive to its victims, who develop a self-destructive lassitude that prevents them from fleeing the region. When a colour has absorbed enough life to grow to full maturity (usually signified by reaching 25 to 30 Hit Dice), it gathers its strength and erupts from its den, coruscating into the sky as it launches the majority of itself back into space. Sometimes, enough remains of the parent colour to survive on its own, and in these cases the life cycle repeats again and again.

Immortal Ichors, which are made of the blood from dead gods. I would say that all oozes with Regeneration, like the Ichor cannot die from old age at all, as the ability simply prevents their death.

Carnivorous blobs could possibly live for centuries, as they are able to hibernate if without food, and ride meteors all the way from Eox to Golarion. There isn't much info about them though.

Carnivorous blobs are building-sized, translucent, crimson monstrosities. Always hungry, they fall into a state of hibernation if they go more than a day without feed on flesh. Some speculate that they are the creation of an insane wizard, while others believe them to be of extraplanetary origin. This latter theory is bolstered by the fact that carnivorous blobs have been spotted hitching a ride on meteors and on the dead planet of Eox.

Oblivions are also said to live for a very ling time, as they usually kill all life in a planet taking centuries to do so.

Oblivions unleashed upon mortal worlds will lay waste to them until destroyed. They often need decades or centuries to end a single world, and are very thorough in their job: their victims are quickly forgotten by others. Some gods claim that oblivions are intended to be released upon the multiverse to cleanse creation and begin a new cycle of life, while others insist their work is nearly complete, leaving only isolated specks of life in the void.

Radiant essences can live for centuries as well:

Radiant essences are mindless and driven to destroy evil to the point of losing all self-preservation. The rare radiant essences that are not destroyed in battle can live for centuries, until they run out of holy energy. The death of an old radiant essence has never been observed.

Deathtrap Oozes may stay indefinitelly in their trap forms:

These oozes surely have an origin rooted in magical meddling. Countless ancient crypts have such oozes within, lurking patiently in ageless trap form, awaiting would-be tomb-raiders for a first-in-centuries breakfast. Those who have studied the deathtrap ooze suppose that wizards of some lost culture altered ochre jellies into perfect guardians.

Finally, Vorgozen, the kaiju ooze, is still new, though her age is actually uncertain. But she is a kaiju, and like all of them, are ageless.

Most kaiju are ancient creatures that have dwelled in hidden lairs since the dawn of civilization, prehistoric monsters woken from eons-long slumbers, or alien entities who visit the world only rarely.

She also has the ability to hibernate:

she can remain infused in the terrain indefinitely, during which time she does not need to breathe, eat, or sleep

Finally, her vague story mentions an "once-flying city", which is a direct reference to the Shory. Their last flying city was Ulduvai, which is the focus of the Crucible of Chaos adventure module, and having ran the adventure, I know the city fell over 5 thousand years ago, making the kaiju at least that old.

But not oozes in general

Other than those with regeneration, there ia nothing in the Ooze creature type that states they are immortal, ageless or imune to dying from old age.