The one intended showstopper in “The Fall of Heaven” is the account of the disappearance of Imam Musa Sadr, the charismatic and revered Iranian-Lebanese cleric. Sadr vanished during a visit to Libya in 1978, and rival theories have competed since to explain his death, most of them implicating Col. Muammar Qaddafi or various Palestinian factions. But Cooper points to Khomeini and his coterie. “The founders of the Islamic Republic were complicit in his murder,” and were said to be intent on sabotaging a secret plan, hatched between the shah and Sadr, to bring the cleric back to Iran as an alternative to Khomeini.

Cooper’s recounting of this scheme is entertaining, but there is scant evidence that Sadr could have formed a serious counterpoint to Khomeini. Most far-fetched is the claim that Sadr was singularly poised to “reconcile faith with modernity.” Sadr’s legend is a beguiling one; ask many in the Middle East about him today, and they’ll still get a faraway, wistful look in their eyes. It’s fanciful, however, to imagine that he could have recast the course of Iranian and Shia history.

But Cooper’s main objective is to rehabilitate the shah, who in the late 1970s became associated with brutality on a scale all out of proportion to the truth. This was largely the work of President Jimmy Carter, who grew so obsessed with the shah’s human rights record that his ambassador to the United Nations likened the shah to Adolf Eichmann. Compared with Iraq’s Saddam Hussein or Syria’s Hafez and Bashir al-­Assad, who massacred in the tens of thousands, Cooper insists “the shah was a benevolent autocrat.” (Carter’s antagonism to the shah coincided roughly with Iran’s inflexibility on oil prices, but Cooper, whose previous book concerned oil politics, oddly makes little of this.)

The numbers of the shah’s victims were far more modest than what Carter claimed — or Khomeini for that matter. The Islamic Republic set about memorializing those victims, and the lead researcher, the seminarian-turned-dissident Emad al-Din Baghi, discovered that instead of 100,000 alleged deaths at the hands of the shah, only a few hundred names could be found.

Nonetheless, if Cooper is going to stake so much on accuracy with numbers, he might have been more scrupulous himself. He mentions no source for his own figure of 12,000 deaths in Khomeini’s first decade and recycles widely rejected figures for deaths in the Iran-Iraq war.

The best-drawn portrait that emerges in “The Fall of Heaven” is of Farah Diba, the shah’s third wife and a figure desperately deserving of a proper biography. Trained as an architect, Farah was educated, cosmopolitan and ambitious, a first lady unlike any the Middle East had ever seen and, in Cooper’s words, “the most accomplished female sovereign of the 20th century.” She rescued lepers, bought Warhols, built museums and turned Tehran into a global hub of artistic and cultural activity. Down to earth, compassionate and clever, she could connect as easily to a cleaner as a courtier, and was the real star of the Pahlavi family.

In passing, Cooper makes the important point that Westerners were rather too fascinated with Khomeini when he first came to power. He “appeared like a mirage out of the Arabian desert with his flowing beard and black eyes to regale them with tales of the bestial Pahlavis.” The French philosopher Michel Foucault traveled to Iran and was infatuated with the spiritual quality of Islamist politics. Journalists “hung on Khomeini’s every word.”

In the light of what we now know, ­Cooper asks us to revisit our inherited memory of the shah, and consider returning with a different verdict. There seems to be lurking in these pages a wish that the shah had cracked down, and kept the forces that opposed him, “the floodgates to today’s carnage,” at bay. But this desire is fundamentally at odds with the personality of the shah, a proud man who rejected the scale of violence the moment seemingly demanded of him. In exile, shortly before his death, a friend asked him why he didn’t crush Khomeini. “I wasn’t this man,” the shah replies. “If you wanted someone to kill people you had to find somebody else.”