R. Aryeh Kaplan's position on the age of the universe has become quite famous over the years. R. Yitzchok Adlerstein summarized it in a Fall 1991 article in Jewish Action. It was subsequently published in a posthumous 1993 book Immortality, Resurrection and the Age of the Universe: A Kabbalistic View. A transcript of a lecture by R. Kaplan on this subject is available for free download here. R. Natan Slifkin discusses this position in his The Science of Torah on pages 115-118 and The Challenge of Creation in chapter 12.
In brief, R. Aryeh Kaplan quotes rishonim who understand literally a midrash that there are 7 cycles to the world of 7,000 years each. This is quoted by the Ramban and Rabbenu Bachya. R. Kaplan then quotes R. Yitzchak of Acre, a student of the Ramban, who writes that each of those years is a year of God. And, he writes, a day of God is 1,000 years, which means that a year of God is approximately 365,250 of our years. Thus, the duration of the universe is 7*7,000*365,250 = 17,897,250,000 or almost 18 billion years.
R. Kaplan writes that according to the view of the kabbalistic work Livnas Ha-Sapir, we are currently in the seventh cycle, which means that the creation discussed in Genesis happened after 42,000 years, which translates into an age of the universe of approximately 15 billion years at "creation" time.
R. Ari Kahn raises a number of question on R. Kaplan's position (see a post to the Avodah list, excerpted from his book Explorations). R. Kahn's objections are as follows:
1. According to Livnas Ha-Sapir, we are currently in the sixth (not seventh) cycle. But the Livnas Ha-Sapir does not mention the "day of God" aspect. According to R. Yitzchak of Acre, we are currently in the second cycle. That significantly diminishes the age of the universe.
2. The Arizal disputed this entire approach and understand the cycles to be spiritual and not physical. According to the Arizal, there is no basis for this entire approach.
The first objection is certainly correct: R. Yitzchak of Acre believed that we are in the second cycle. That would mean that he was of the view that the universe was slightly over 2.5 billion years old at "creation" (i.e. the creation described in Gen. 1). However, and R. Kahn certainly agrees to this, R. Yitzchak of Acre believed that the universe is billions of years old! That, in itself, is significant.
I'm not sure about the second objection. R. Aryeh Kaplan specifically raised the issue of the Arizal's view and denied its overriding significance (Immortality, Resurrection and the Age of the Universe, pp. 6-7):
Before going any further, it must be mentioned that most of the more recent Kabbalistic texts do not make any reference to these teachings. This is because two of the greatest Kabbalists, Rabbi Moses Cordovero (the RaMaK) and Rabbi Isaac Luria (the Ari) disputed this concept in general...That is presumably why the Tiferes Yisrael and Rav Kook felt comfortable citing this position. In a very recent article in the Torah U-Mada Journal on this doctrine of Sabbatical years, Raphael Shuchat suggests (here - PDF, n. 99):
Here, however, the second principle that was discussed earlier comes into play. Since this is not a matter of law, there is no binding opinion. Although the Ari may have been the greatest of Kabbalists, his opinion on this matter is by no means absolutely binding. Since there were many important Kabbalists who upheld the concept of Sabbatical cycles, it is a valid, acceptable opinion.
Despite the fact the R. Kook follows the Ari in most areas of Kabbalah, he felt that an idea found in the writings of Ramban and R. Bahya remains legitimate despite the Ari’s criticism.If Rav Kook felt that this position is kabbalistically justifiable, I certainly won't criticize it. Thus, while R. Aryeh Kaplan's approach might not be consistent with the most current view of the age of the universe, it demonstrates that a medieval kabbalist believed that the universe was greater than 2 billion years old, which offers justification for those in the contemporary society who believe that the universe is older than 5,766 years.