There's no backup in the sunnah of the prophet() for doing this.
And scholars consider repeating the recitation of al-Fatihah as makrooh to a level that it could invalidate the prayer, for details see Can you read sura Fatihah after the Fatihah in Salaah?.
The only statement about repeating the recitation of the same surah in the same raka'a in fiqh books is that of the maliki scholar as-Sawi أحمد الصاوي in his a-Sharh as-Saghir (the small explantion) called Bulghat as-Saalik liAqrabi al-Masaalik بلغة السالك لأقرب المسالك (this is a marginalia on a commentary of the book of sheikh ad-Dardir called Aqrab al-Masalik limadhhabi al-Imam Malik أقرب المسالك لمذهب الإمام مالك) saying:
In the following I will translate from Arabic as these translations are of my own take them carefully
وَقَدْ وَرَدَ عَنْ مَالِكٍ كَرَاهَةُ تَكْرِيرِ السُّورَةِ كَالصَّمَدِيَّةِ فِي الرَّكْعَةِ وَظَاهِرُ مَا وَرَدَ عَنْ مَالِكٍ الْكَرَاهَةُ وَلَوْ فِي النَّفْلِ ، وَهُوَ خِلَافُ مَا فِي كَثِيرٍ مِنْ الْفَوَائِدِ ، وَلِذَلِكَ سَيَأْتِي فِي الشَّرْحِ الْجَوَازُ فِي النَّفْلِ. (Source Fatwa islamweb #133229)
It was narrated from Malik that he considered repeating a surah like the Samadiyah (I guess he refers to surat al-Ikhlass) in the same raka'a as makrooh (disliked). And the apparent what was narrated from Malik was the dislike even in nafl (optional) prayer, and this is agaiant many benefits, that's why we will explain with details the permissibility for optional prayers later.
Note that Malikis consider such a repetition even in two different rak'as as disliked as stated in the same fatwa, but there's a hadith showing that this was practiced by the prophet () in Sunan abi Dawod and in the Sahih books one may find the prophet() confirming this for a not named sahabi (see for example in Sahih Muslim). Nevertheless doing this in all prayers with the same surah or believing it is sunnah or rejecting to pray with other surahs (especially if one pretends it is not permissible) is an issue and makrooh in all madhhabs.
In the same fatwa they say it is stronger view قول متجه to say that this (the dislike of repeating the same surah after al-Fatihah in the same raka'a) is against the sunnah due to the fact that it was not mentioned to have been practiced or even approved by the prophet () or the sahabah.
The only hadith I have in mind that might be used as an evidence to support this is:
A man entered the mosque and started praying while Allah's Messenger (ﷺ) was sitting somewhere in the mosque. Then (after finishing the prayer) the man came to the Prophet (ﷺ) and greeted him. The Prophet (ﷺ) said to him, "Go back and pray, for you have not prayed. The man went back, and having prayed, he came and greeted the Prophet. The Prophet (ﷺ) after returning his greetings said, "Go back and pray, for you did not pray." On the third time the man said, "(O Allah's Messenger (ﷺ)!) teach me (how to pray)." The Prophet said, "When you get up for the prayer, perform the ablution properly and then face the Qibla and say Takbir (Allahu Akbar), and then recite of what you know of the Qur'an, and then bow, and remain in this state till you feel at rest in bowing, and then raise your head and stand straight; and then prostrate till you feel at rest in prostration, and then sit up till you feel at rest while sitting; and then prostrate again till you feel at rest in prostration; and then get up and stand straight, and do all this in all your prayers." (Sahih al-Bukhari)
But it is still a weak evidence as he () didn't say you may repeat it. As a repetition wouldn't make sense.
Note that scholars lax the rulings for optional prayers therefore such a repetition might be allowed for some benefits like memorization.
So simply speaking it is not recommended to do this however there's no backup or scholarly view saying that repeating the recitation of a surah in the same raka'a unless it was al-Fatihah would invalidate the prayer.