Asimov was working off a population model proposed by Malthus in 1798 [0], which has long been known to be flawed. It's not just industrial human populations that have slowing growth rates; most populations creatures have slowing growth rates as they near the population level the environment supports. The most common model for this is called "logistic growth" [1].
The "current doubling time" is kind of a misnomer -- it's how long it would take the population to double given the current growth rate, but the rate itself is also shrinking. Human population growth reached its peak (percentage-wise) in the early 1960s [2] and most credible estimates now have the world population leveling off in the 9-12 billion range (less than a full doubling from current levels) [3].
