During the last three major geological epochs-- the Miocene, Pliocene, and Pleistocene-- we see periods of global cooling that correlate quite closely with what appear to be significant changes in anatomical characteristics of primates, eventually leading to the rise of bipedalism and the human species.

In the middle Miocene, cooling temperatures seem to be associated with the split of apes from monkeys in Africa; larger bodied primates who were better adapted to descend from the increasingly sparse tree canopy in some regions, spending more time on the ground between patches of forest.

In the late Miocene, we again see global temperatures decreasing, and dryer conditions leading to grassland environments spreading in many parts of the world, including Africa. And by sometime around 12-10 million years ago, we first see the split between gorillas' and the chimpanzees' ancestors, and then maybe around about 7-8 million years ago, the split between the other chimpanzees and our earliest semi-bipedal ancestors.

The theory goes, again, that receding forests led to increased time spent out of the trees, and that in that environment / circumstance, the ability to walk bipedally-- first for short periods, then gradually favored by more significant anatomical changes-- came to represent a significant survival benefit. By 5 million years ago, you have bipedalism, by 3.5 million years ago, you have Australopithecus afarensis, and by 1.8 million years ago, you're into the Homo line.

Most of this seems to be happening as global climates and environments were shifting away from the more tropical, wetter environments that characterized the world through much of the Eocene and Oligocene and the early Miocene, and shifting to dryer and more open environments in regions where large-bodied primates were living.

Anatomically modern humans and Neanderthals also arose during periods of increased glaciation that occurred during the Pleistocene.

In more recent periods-- during the Holocene-- climate change and climate event have generally been less pronounced. More problematically, a lot of the record of adaptations to those climate events is under water. In the Southeastern US, for example, there are huge swaths of land in the Gulf of Mexico that would have been exposed during the last glacial maximum. Thousands of years of human history are likely submerged on the continental shelf, making it very difficult to assess how people dealt with climate change and rising seas (aside from moving inland, which they seem to have done).

There've been arguments that climate events like the 8.2 kiloyear event and other, similar Bond events (which are thought to occur on a roughly 1500 year cycle) can be implicated in various periods of cultural upheaval, but the problem is that these climate events are difficult to identify, and even harder to tie to changes in cultural practice.

We can point to periods like the medieval warm period, or the little ice age, when global temperatures fluctuated rapidly and relatively severely, but by that point, it's clear from written records what was going on, so I think that sort of steps beyond your question.

Human societies would have recognized that climates were changing. Without written records, oral histories were kept and recounted. Memories would be tied to places and times during the year when groups occupied / visited those places. When sea levels rose, stories of locations that were once accessible and became inaccessible due to rising waters would have been a common occurrence in coastal regions. Recent work with Australian oral traditions suggests that there's at least a 7000 year record of sea level rise and the effects of climate change preserved in those stories. It's likely that groups occupying coastal regions elsewhere in the world would have maintained similar oral histories.

More specific follow up: How did periods of warming/cooling affect culture, lifestyle, and religion?