The company is best known for creating family-friendly holidays, but the facilities at Casa Cook are focused on grown-ups. Drifting through the resort is the constant hum of electronic dance music. There is a well-equipped gym, an outdoor yoga deck and Turkish-style hammam - but no play areas. One morning, I see a child entertaining herself by building a pile of rocks. Another kid balances a plum on his knuckle to pass the time. Around them, adults are piling into a nutritious buffet breakfast at the open-plan restaurant. Dried coti berries and pumpkin seeds are available; sausages are not.
The first Casa Cook resort opened in Rhodes last year and Fankhauser wants to create 10 more over the next three years. Only two locations, in Croatia and Crete, are so far confirmed. He concedes that there will never be a huge number of Casa Cook resorts: "It's not going to be 100 because... we don't want to dump the market."
These hotels are intended to be Thomas Cook's high-end offering but Fankhauser admits they may represent just "1 per cent" of the overall business. So, is Casa Cook just a very plush marketing gimmick?
He insists the resorts are part of a broader strategy. Five years ago, the company offered packaged stays at more than 10,000 properties, the majority of which were owned and operated by outside hoteliers. At most of them, Thomas Cook would merely pre-book a number of rooms for its customers, who would be holidaying alongside the guests of other tour operators.
Today, it offers just 2,500 hotels. The company owns five of these and just over 180 are "own brand" properties - including Casa Cook - which are run on management contracts, in which Thomas Cook has full control over the look and feel of the hotel. The rest are "selected partner" hotels, with which Thomas Cook pays to gain exclusive access or a large allocation of rooms. In return, it demands a decisive say over how each place is run - everything from deciding on what is on the restaurant menu to how maids do the bedding.
Slashing the number of hotels means it can streamline the business, reducing, for example, the number of holiday reps and safety auditors, while maintaining tighter quality control. "You want to do something like this where the margin is much higher," says Fankhauser. "You essentially make more money from operating a hotel like this."
And though bohemians want to tread an unworn path, Fankhauser says there are benefits to package holidays that will ensure that they remain popular. Two days after my visit, an earthquake hit Kos, killing two tourists and injuring hundreds. Casa Cook suffered a short power cut, the guests were unharmed and all continued their stay at the resort. But some Thomas Cook customers elsewhere on the island were moved or flown home. "If something goes terribly wrong, we are there," says Fankhauser. "That is a form of reassurance and we are not shy to talk about it."
On one wrist Fankhauser wears an expensive-looking silver Omega, on the other a Fitbit fitness tracker. He has worked in the travel industry for 28 years, over which time he has gained a reputation for turning around failing companies, and joined Thomas Cook in 2001 as it faced a turbulent decade.