Deep time means that the Earth and the Universe are billions of years old. Currently, Scientists estimate that the Solar System is about 4.5 billion years old and that the Universe is about 13.76 billion years old (Ages of the Earth and the Universe).
Speculation before modern times
Though there were many premodern speculations about an old or eternal Universe, they usually involved a Universe much like its present state. Hindu beliefs are typical, and these beliefs have inspired some Creationist-style Pseudoscientific works. Michael A. Cremo and Richard L. Thompson have claimed that our species has existed for hundreds of millions of years in their books Forbidden Archaeology and The Hidden History of the Human Race (Hidden History, Hidden Agenda). The second book is an abridged version of the first one, and that one was published by the Bhaktivedanta Institute in San Diego, complete with being dedicated to "His Divine Grace, A. C. Bhaktivedanta Swami Prabhupada".
The last few centuries
The idea of an old but very different universe emerged over the last few centuries, notably with the mapping out of geological time in the early nineteenth century. Most of the familiar era and period names date back to then. It became evident that the Earth's inhabitants looked more different the farther back one goes and that our earliest ancestors were Apes, with humanity being an extreme latecomer.
The Universe turned out to be about 3 times older than the Earth, and its early stages very hot and compressed (the Big Bang), another departure from its present state.