Why do we say היום הרת עולם on rosh hashana if the world was created five days before on the 25th of elul?
The second cheilek of Talelei Oros on Rosh Hashanah (p.קסד) notes how the Avudraham explains that the term הרת is used to represent the "pregnancy" of the world coming to its conclusion.
We say in Yirmiyah 20:17:
וְרַחְמָ֖הֿ הֲרַ֥ת עוֹלָֽם
And her womb big [with me] for all time
The inference being that the world's creation was its "pregnancy" - and the term 'pregnancy' is also used to denote the time of birth. Like we say in Iyov 3:3
A male has been conceived!
Likewise refer to Rosh Hashanah 27a:
כְּמַאן מְצַלֵּינַן הָאִידָּנָא ״זֶה הַיּוֹם תְּחִלַּת מַעֲשֶׂיךָ זִכָּרוֹן לְיוֹם רִאשׁוֹן״, כְּמַאן — כְּרַבִּי אֱלִיעֶזֶר, דְּאָמַר: בְּתִשְׁרִי נִבְרָא הָעוֹלָם
In accordance with whose opinion do we pray today on Rosh HaShana, saying: This day is the beginning of Your works, a commemoration of the first day? In accordance with whom? In accordance with the opinion of Rabbi Eliezer, who said: The world was created in the month of Tishrei.
Therefore, on the sixth day, when Adam was created, we celebrate Rosh Hashanah as it marked the day when the "pregnancy" of the world came to an end i.e. man's creation marked the purpose of creation, and was thus fitting to be judged.
The Hebrew expression הרת עולם is better translated into English as "the conception of the universe/world", not the pregnancy of the universe/world, like עולם מעוברת, from the root עובר.
To be pregnant is to be with the developing potential child, called עובר. Conception, however, refers to the beginning moment of fertilization. When the innovation of potential new life originates.
By saying the Conception of the Universe in association with the creation of Adam HaRishon, it emphasizes the relationship between the two.
That in G-d's act of creating Adam HaRishon (the Human species), was embedded the purpose and intention behind the creation of the entire universe.
That within the species of HaAdam is found the paradigm for the structure and form of the entire creation which preceded it.
In other words, the human species is the Microcosmos (עולם קטן) in relation to the Macrocosmos of the entire Universe like it states in Midrash Tanchuma, Pekudei 3:3, Duties of the Heart, Second Treatise on Examination, Chapter 5.
And this is in keeping with the saying of our Sages (Noam Elimelech, Sefer BaMidbar, BaMidbar 1:4), "The final act arose in thought to begin with." סוף מעשה עלה במחשבה תחילה
That the act of the creation of the Human Species was conceived and intended in the beginning of the first thought of the entire Creation. The Human species is the ultimate purpose and final intention of the entire Creation.
This is the meaning of the first word that begins the Torah, בראשית, like Rashi explains. That Bereshit means, for the sake of Reishit, G-d created the Heavens and the Earth. That Reishit refers to the Torah and to Israel.