I ran into this word in a well-written article by Natalie Angier. Since I like to learn new words, I looked for the definition of pericentury. But was unable to find it easily. Fortnately, I was able to track it down.
Peri as a prefix means around. Therefore pericentury means around a century. This definition makes sense to me since about a 100 years is the maximum human lifespan. Therefore, in the context of the article, you have a pericentury of consciousness alloted to you.