“The Cottage,” which opened Monday at Broadway’s Hayes Theater, is a new comedy by Sandy Rustin that already feels dated — a slamming-door farce without a lot of doors and a cast of randy characters who spend most of their time mugging at the audience while struggling to maintain their plummy British accents. The biggest laughs come from Paul Tate dePoo III’s spectacular scenic design, which not only re-creates a lavish 1920s-era country estate but also embeds it with a bounty of sight gags, most coming in the form of cigarette holders and lighters that are cleverly hidden in random objects like stair knobs, silver tea sets and a foot-tall replica of Michelangelo’s David.

It’s the sort of show where the entrance of a new character is greeted with a freeze-frame moment where the other castmates all stop in place, the lights dim, a spotlight centers on the new arrival and a musical flourish adds just that extra bit of overkill.

Laura Bell Bundy, who’s surprisingly been a stranger to Broadway since her debut in the musical Legally Blonde 15 years ago, vamps it up as Sylvia, who’s just finished her annual one-night dalliance with her brother-in-law, Beau (Eric McCormack, mostly playing things straight), when she decides to make the clandestine romance official with a series of ill-timed telegrams. Soon we learn that Beau’s very-pregnant wife (Lilli Cooper) and her husband, Beau’s brother Clarke (SNL alum Alex Moffat), are eager to end their marriages and make their own even-more-established affair official. But Beau is less than eager for a simple wife swap — especially when his more regular mistress (Dana Steingold, delightfully goofy) turns up, with her murderously jealous ex-husband (Nehal Joshi) following soon behind.

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Eric McCormack, Laura Bell Bundy, Alex Moffat), Lilli Cooper and Dana Steingold in “The Cottage” (Photo: Joan Marcus)

The star-filled cast can best be described as game, deserving the tip of a ’20s flapper hat for managing to keep the nonsensical story afloat for as long as they do. A special shout-out to Bundy, Steingold and especially Moffat, a Broadway newbie who leans into the physical comedy in a way that steals attention even when he’s merely sitting down (in a deliberately awkward and exaggerated way).

There’s nothing subtle about either the script or the performances under director Seinfeld alum Jason Alexander’s direction, and alas, there’s nothing particularly funny here either. The plot is utter nonsense, the sort of piffle that collapses on even the most rudimentary contemplation, and even some of the biggest twists have been flagged far, far in advance, as if by semaphore.

More alarmingly, the laughs are surprisingly sparse. By the second act, when dePoo’s set has been depleted of sight gags, the comedy has clearly run out of gas and the faulty mechanics of the plot begin to heave under the strain. Naturally, this is when Rustin stoops to a prolonged fart joke. Flatulence is a sure sign that the comedy’s gone flat.