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Putting on layers of clothing, building insulated shelters or simply lighting a fire are some simple yet effective ways of staying warm. Humans have shown they are quite adept at keeping themselves warm, even surviving through the last major ice age some 11,700 years ago.

Eventually, the Earth started to warm up and humans found themselves nearly everywhere. When you think of the hottest places on Earth, you would picture a desert. The largest of the hot deserts is the Sahara with average temperatures reaching around 45°C or 113°F. The Sahara is so large it covers 10 countries, one of the most ancient being Egypt which has been around since 3000 BC. As seen below, there are depictions of servants wielding fans, though most likely only the nobility would have this luxury. Undoubtedly Egypt was warm and it would be a few thousand years before the invention of electric fans and air condition. So how did everyone in those times fight the heat and stay cool?

Homo sapiens evolved near the equator so have always had to deal with hot temperatures. There are a lot of articles about how people stayed cool in the days before air conditioning:

1. Staying out of the sun at the hottest periods and possibly napping or otherwise reducing activity (the siesta in Spain)


2. Buildings with thick walls to stay cool all year round (stone or mud will do this)


3. Building techniques to promote shade such as verandas, porches, balconies, brise-soleils, porticos, shady courtyards, walls, trees, etc (small windows/not building entirely out of glass also helps)


4. Buildings designed to promote ventilation, with high ceilings, routes for air to blow through, wind towers and wind catchers (towers designed to channel cool air to the ground); and sitting on the porch also exposes you to winds


5. Water, not only to drink but spread on roofs, on curtains, etc, to cool as it evaporates


6. Dressing appropriately (often long loose clothes, generally white)

Sources:

• Accu Weather: 5 ways people stayed cool before air conditioning was invented
  


• Forbes: How Did Humans Handle Intense Heat Before Air Conditioning Became Popular?
  


• Apartment Therapy: How Did People Survive Before Air Conditioning?
  


• The National (UAE): Traditional ways to keep cool during summer
  

SHORT ANSWER

Ancient Egyptians kept cool by using, among other things,

• Mud brick walls to keep houses cool


• Windows for cross draft


• Vents in the roof to help air circulate


• Matting for window shades


• Damp reeds and water pots for evaporated water to cool the air


• Roofs to sleep on at night


• Simple, light clothing, but clothing was also dictated by fashion and status


• A headrest at night to improve air-circulation round the head


• A wet blanket to sleep in


• Tree-lined paths

Note: The Ancient Egyptians mostly lived along the Nile rather than in the desert which, as today, was largely uninhabited.

HOUSING

Mud bricks were commonly used to build homes:

The sun-dried mud bricks, known as djebat, were used to build walls
that were then covered with mud plaster and decorated in white or soft
colors. The brick walls were well suited to Egypt’s climate as they
kept homes cool in the summer and warm in the winter.

Further,

Windows were small squares, set high up in order to mitigate the
stifling summer heat and prevent dust, glare and flies from entering
the house. Windows were arranged opposite each other to allow a cross
draft or breeze to filter through; vents in the roof also permitted
air to circulate. Loosely woven matting was used as a window shade to
stifle the heat and the glare of the sun.

Water was also used:

...people hung damp reeds over windows and placed water-filled pots in
hallways. As the water evaporated, it would cool the air.

The poorer class

lived in one-room huts under roofs made of reeds, straw and grass.
Roofs also served as sleeping quarters on hot summer nights...

Nights can actually get fairly chilly in and around the desert (when travelling in Mali on the edge of the Sahara, I slept on rooftops a couple of times; by midnight, a blanket was needed).

CLOTHING

Although fashion and status also determined the way people dressed,

In accordance with the temperature, the Egyptians devised simple
styles and comfortable materials in which to dress from the earliest
eras. Cotton was a major crop put to good use, and linen, especially
the special material called BYSSUS, became the basis for clothing for
upper classes.

Source: Margaret Bunson, Encyclopedia of Ancient Egypt

The masses mostly wore light garments as the last you want in a hot climate is anything heavy. Clothing was simple for the poor:

During the hot summer months, slaves, reed gatherers, brickmakers,
fishermen, boatmen and children wore little or no clothing at all.
Nudity was not a problem, in those days.

Linen was the fabric of choice, because of its light, airy quality and
ease in mobility.

A piece of cloth was sometimes used to cover the head to protect against the sun.

Source: Ancient Egyptian Peasants

OTHER

Ancient Egyptians also used a headrest at night to help keep the head cool:

The use of a headrest not only ensured a relatively comfortable
position while sleeping, but also enabled air to circulate around the
head, an added advantage in a warm climate like that of Egypt.

Some headrests were made of wood, but the poor often used stone. These were also used in other regions around the Sahara:

In neighbouring Libya headrests were also found among the Garamantes
who ruled the Fezzan between 500 BC and AD 700. In Mali, they were
present among the Tellem, from the eleventh to fourteenth centuries
AD, and...Dogons, until the
twentieth century. Nineteenth-century headrests belonging to the
Bicharin and the Ababds in Nubia still present forms similar to those
of their ancient counterparts,...

"Ancient Egyptian Carved Wood Headrest (1550 BC to 1186 BC". Source: Finch & Co.

A number of websites, such as this one, also note the use of a wet blanket to stay cool at night. People

wrapped the blankets around their bodies and went to sleep. Throughout
the night, the heat caused the water to evaporate off the blankets,
keeping the person inside nice and cool.

Ancient Egyptians and others living in and around the desert would also undoubtedly have done many of the things which their modern counterparts do, such as cooling off in the Nile (or the Niger etc.) and taking advantage of whatever shade there was from trees (wealthy Egyptians sometimes had fine gardens with tree-lined paths and running water). Even so, religious ceremonies were often held out in the sun, and Egyptians were evidently not unused to it:

Many ceremonies were held in the open sunlight, a custom that brought
about complaints from foreign dignitaries. These ambassadors and
legates from other lands attended the ceremonies in honor of Aten and
suffered heatstrokes as a result.

Source: Bunson

(all highlighting is mine)

Other source:

A. R. David, The Pyramid Builders of Ancient Egypt