It was at number seven on The Sunday Times bestsellers list last week.
The critically acclaimed book by a former Financial Times Moscow correspondent faced an onslaught of law suits from Russian oligarchs including Abramovich, three of whom have now been targeted by UK sanctions.
The book was a Sunday Times bestseller when it first came out and sales have spiked again as the war in Ukraine focuses interest on Russia's leader.
"I wish the circumstances were different. But I'm very grateful for all the attention to Putin's People," Belton tweeted.
She thanked publisher Harper Collins "for making sure that it remained on the bookshelves despite an unprecedented barrage of lawsuits last year".
The publisher eventually settled out of court in December last year with the main claimant Abramovich.
It agreed to alter some passages in future editions of the book about him, including over the Kremlin's role in his acquisition of Chelsea football club.
Belton spoke to the House of Commons Foreign Affairs Committee on Tuesday at a discussion on the use of such lawsuits to hinder public participation in politics.
"No matter how good the sourcing is on some of these claims, and how great the public interest is, the cases are just too expensive to defend, the system is stacked in favour of deep-pocketed litigants from the outset," she told the committee.
"Quite frankly when we were facing it at the beginning, I didn't know whether the publisher would be able to withstand the barrage of claims," she said.
"I mean this is five big cases from some of the world's richest men all at the same time, and I didn't know whether we would have to withdraw the book."
Abramovich, who bought Chelsea in 2003, was placed under UK sanctions on March 10, including an asset freeze, a travel ban and a ban on transactions with UK individuals and businesses.
These sanctions mean he would now not be able to launch such legal action at the High Court using a British law firm.
The UK has altered legislation to make it easier to impose sanctions on oligarchs who have already been placed under sanctions by ally countries.