The Machatzis Hashekel (O.C. 417) quotes the Hagahos Mordechai in the name of the Tosefta that there is no Rosh Chodesh Bentching for the month of Av. Many poskim argue and hold that we do bentch Rosh Chodesh on Shabbos Rosh Chodesh Av. The minhag is to bentch Chodesh Menachem Av, even though the month is called Av. We say it so that we should be zoche to the geulah (Aruch Hashulchan E.H. 126).
The Tchebiner Rov zt”l was once asked how to date and documents written in the month of Av. He said that the answer can be found in the Teshuvos Chasam Sofer, Orach Chaim.
The questioner came back to the Tchebiner Rav and told him they searched the whole sefer and couldn’t find any such discussion. He told them, “When learning a Teshuvas Chasam Sofer, it is not enough to learn the text, but one needs to learn the date as well.”
If you check Teshuvos # 32, 33, 35, 37, The ones written before Tisha B’av the Chasam Sofer dates it as Av. The ones written after Tisha B’av he dates them Menachem or Menachem Av.
(The Chasam Sofer himself writes Befeirush in the Hagahos on Ch. 551, also in the Drashos 2: p331. “After 9th Av a new month starts called Menachem.”)
I leafed through the seven volumes of Shu”t Ch. S. (1246 Teshuvos) and found, there are aproximately 150 Teshuvous written by the Chasam Sofer in the month of Av. All of them follow the rule of the TchebinerRav zt”l, The ones dated Before 9th Av are dated as Av. The ones after 9th Av are dated as Menachem. However there are two days, the 13th and 15th of Av where the Ch. S. dates them Av, even though they are after 9th Av.
It is possible the 15th of Av being it is mentioned in the Mishna (end m. Ta’anis) as Chamisha Asar B’av, The Ch. S. didn’t want to change from the Loshon of the Mishna*.
On the 13th of Av the only explanation I can think of, being it is the Yahrzeit of his first Rebbetzin שרה.
15th of Av 13th of Av
O.Ch. 60 O.Ch. 115
Y.D. 129 O.Ch. 158
E.H. 2: 72 Y.D. 43
Chelek 6 41
Chelek 7 18
*There is one Teshuva (E.H. 2 : 125) on the 15th he dated as Menachem.