Jeg åbner vinduet for at køle rummet ned.

Because åbner is the present tense form of the verb at åbne (to open).

• at åbne = to open
• jeg åbner = I open / I am opening

In Danish, the present tense usually ends in -r for many verbs.

Examples:

• jeg åbner = I open
• du åbner = you open
• han åbner = he opens

Unlike English, the verb does not change for different persons in the present tense, except that the same form is used with all subjects.

It can mean either, depending on context.

The Danish present tense often covers both:

• I open the window
• I am opening the window

So Jeg åbner vinduet could describe:

• a general action,
• something happening right now,
• or sometimes a near-future action, depending on the situation.

Danish does not usually need a separate form like English am opening.

Because vinduet means the window, while vindue means window.

Danish often puts the definite article at the end of the noun instead of using a separate word like English the.

• et vindue = a window
• vinduet = the window

So in this sentence:

• Jeg åbner vinduet = I open the window

This is very common in Danish.

Because Danish usually expresses the with a suffix attached to the noun.

Here:

• vinduet = the window
• rummet = the room

Both vindue and rum are neuter nouns, so their definite singular form ends in -et.

Compare:

• et vindue = a window
• vinduet = the window
• et rum = a room
• rummet = the room

So the meaning of the is already built into the noun.

for at often means to or in order to when expressing purpose.

So:

• Jeg åbner vinduet for at køle rummet ned means
• I open the window in order to cool the room down

It tells you why the speaker opens the window.

A useful way to think of it:

• for at = in order to

In many cases, English just uses to, while Danish often uses for at more explicitly when talking about purpose.

In this sentence, for at is the natural choice because it clearly expresses purpose.

• for at køle rummet ned = in order to cool the room down

Using only at here would sound wrong or incomplete in standard Danish.

So after a main clause like Jeg åbner vinduet, if you want to explain the purpose, for at is the normal pattern.

køle ... ned is a very natural everyday Danish way to say cool ... down.

It is a verb + particle construction:

• køle ned = cool down

When there is an object, the object often goes between the verb and the particle:

• køle rummet ned = cool the room down

This works a lot like English:

• cool down the room
• cool the room down

You may also see nedkøle, but that often sounds a bit more formal, technical, or less conversational depending on context. In ordinary speech, køle ... ned is very common.

Because ned is acting as a particle, and in Danish these particles are often written separately.

So:

• køle ned = cool down

When there is a direct object, it commonly goes in the middle:

• køle rummet ned

This is similar to English separable phrasal verbs:

• turn off the light
• cool the room down

So the split is normal Danish grammar, not a mistake.

ned literally means down, and here it gives the idea of lowering the temperature.

So:

• køle = cool
• køle ned = cool down

Without ned, køle can still mean cool, but køle ned strongly emphasizes the change to a lower temperature. It is the most natural choice in a sentence like this.

Because rummet means the room.

The base noun is:

• et rum = a room

The definite form is:

• rummet = the room

Again, this follows the common Danish pattern for neuter nouns:

• et
  • noun in the indefinite singular
• -et added to make it definite

So:

• et rum → rummet

The basic structure is:

Subject + finite verb + object + purpose phrase

So here:

• Jeg = subject
• åbner = finite verb
• vinduet = object
• for at køle rummet ned = purpose phrase

Danish is a verb-second language, which means the finite verb normally comes in the second position in main clauses.

In this sentence, the subject comes first, so the verb comes right after it:

• Jeg åbner ...

Then Danish word order would change because of the verb-second rule.

For example:

• Nu åbner jeg vinduet for at køle rummet ned.
• Now I open / am opening the window to cool the room down.

Notice what happened:

• Nu comes first
• then the finite verb åbner
• then the subject jeg

So Danish does not keep jeg before the verb when something else starts the sentence.

These are two important Danish vowels:

• å in åbner sounds roughly like the vowel in English more or law, though not exactly the same.
• ø in køle has no perfect English equivalent. It is a rounded front vowel. A rough approximation is something like the vowel in French deux or German schön.

Very roughly:

• åbner ≈ OHB-ner
• køle ≈ something like KUH-luh with rounded lips, but this is only approximate

The exact Danish sounds are worth practicing separately, because English does not have perfect matches for them.

Yes, it sounds natural and idiomatic.

A Danish speaker would readily understand:

• Jeg åbner vinduet for at køle rummet ned.

It is a normal way to express:

• opening the window
• with the purpose of making the room cooler

The combination åbne vinduet and køle rummet ned is very ordinary everyday Danish.