Frost passed away from a heart attack at the age of 74 while he was on board the Queen Elizabeth cruise ship last weekend.
Hamilton has claimed that he was "lucky" to get to know Frost, who interviewed the 28-year-old earlier in the summer.
"I was shocked to hear the sad news about Sir David Frost's passing earlier this week," Hamilton wrote in his BBC Sport column.
"He was a great man. His work ethic, will to win and desire to be the best in what he did was inspiring.
"It was a big surprise to hear what had happened. When we'd met he was a little frail but sharp as ever."
Meanwhile, Hamilton is preparing for this Sunday's Italian Grand Prix.