Mortdecai is creeping into theaters with the flushed shame of a debutante who expects to be pelted with tomatoes. It’s a pity. In 1965, Mortdecai would be the hit of the year. Director David Koepp whips through this pop-colored caper about crooked art dealer Charlie Mortdecai (Johnny Depp) — one of those posh British frauds with a large estate and no money in the bank whose existence is, like a faked Rembrandt, merely a good-looking scam — as though he thinks that, if the dialogue speeds fast enough, the entire film can DeLorean back five decades to where it belongs.

Comedy molds quicker than other genres. A kiss or a punch might pack the same smack as the day it was filmed, but a punchline soon rots. Mortdecai feels fusty from the moment it begins. The opening slapstick sequence is an exact mimic of the Chinese nightclub in Indiana Jones and the Temple of Doom, itself a hat-tip to the past. There are vengeful Asian gangsters, red and blue neon, a tense exchange of valuables and a wacky fire battle that winds up broiling the bar. Whenever Depp smacks the table, Koepp cues a dopey sound effect, a gag that hasn’t made adults chuckle since Laugh-In.

If it were unearthed today as a lost spiritual sequel to The Pink Panther, Mortdecai might have been hailed as a treasure. But there’s no room at the modern multiplex for an homage to kooky classic comedy unless we’re snickering at the very idea, as in the Austin Powers films. And for better and worse, Koepp is sincere. He has great affection for this world of overstuffed libraries and leather chairs, a place that looks like it’s from a Tintin comic but whose obsessions are resolutely adult. Mortdecai’s idea of a prank involves tricking someone into drinking cheap port.

The first 10 minutes of the film are devoted to Mortdecai’s newly grown mustache, which is literally the only topic of conversation. He preens it with a tiny comb, smoothing the curls from his nose to the corners of his gummy grin. It’s a rich caramel that matches his gelled hair, his brown suit and the corgi he shares with wife Johanna (Gwyneth Paltrow). She can’t stand the thing. Every time she kisses him, she gags, causing Mortdecai to panic that Johanna will leave him for MI5 agent Martland (Ewan McGregor), who’s pressuring the couple to find a missing Goya that may lead to a Swiss bank stash of Nazi gold.

The best thing about the mustache is that it keeps Depp from wearing a fedora. He never acts without a security blanket from the costume department. In the ’80s, it was teardrop tattoos and scissor hands; then came two decades of hats. Next, perhaps a monocle. But after a string of expensive misses, Depp needs to make sure there will even be a next. His charm has calcified into caricature: He’s played pirate, Mad Hatter and Native American with the same extravagance. Here, Mortdecai simpers like an ingenue, burbles as if he has indigestion and runs in mincing steps with his hands flipped out at the wrists as if he’s dancing the Charleston.

The irony is that Depp’s shtick is ideal for the part — it’s just so similar to his bad roles that we can’t tell he’s doing good work. The Mortdecai of the comedy novels from the ’70s (themselves 10 years too late for popular success) was a cartoonish clod who’d have died on page one if not for his heroic manservant, Jock (Paul Bettany with a fierce facial scar). He was a snob, a drunk and a frustrated lech. The latter two partially explain Mortdecai’s R rating, but his boner jokes are so genteel that most of them will flutter right past the children who have formed the bulk of Depp’s fan base.

The rest of the cast acts at a less obnoxious pitch, allowing Bettany, Paltrow and McGregor — best known for their serious dramas — to prove themselves surprisingly deft comedians. When McGregor’s agent pretends to soothe Mortdecai’s anxiety about his marriage, he adds a lusty emphasis to the line, “Don’t worry about your wife, I’ve been keeping her filled in.” Their performances deserve to be seen. The hard part will be convincing audiences to shake off their Depp fatigue and embrace a film that’s daffy, dated and precisely as intended. If we can’t, then the joke is on us — a gantlet Koepp throws down in a sly, silent gag when Mortdecai flits to Los Angeles and discovers, to his shock, a whole elevator of hipsters with retro mustaches. We’re already living in Mortdecai’s world. We may as well laugh at it.

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