There is a machlokes beis shamai and beis hillel regarding a succah made more than 30 days prior to succos. Beis Shamai holds that it is passul and Beis Hillel holds that it is kasher. The Mishna ends off by saying that if a Succah was built לשם חג, even if built from the beginning of the year it is kasher. Rashi says that the last line of the Mishna is only necessary for Beis Shamai because according to Beis Hillel it is kasher anyway. Tosafos on the other hand cites the Yerushalmi that holds that even Beis Hillel partially admits to Beis Shamai that if built not לשם חג more than 30 days prior to Succos it is necessary to be מחדש בה דבר, meaning to redo some amount of sechach spanning accross the width of the succah. Based on this Tosafos explains that the last line of the Mishna can fit with Beis Hillel as well. The advantage of constructing the Succah לשם חג is that he will not need to be מחדש בה דבר. Rashi seems to understand that the Bavli which makes no mention of this requirement within Beis Hillel would apparently hold that it is unnecessary to do anything. Since Beis Hillel holds that a Succah doesn't need to be built for the purpose of Succos, it would not be necessary to be mechadesh anything in the Succah.
According to Tosafos who holds that within Beis Hillel it would be necessary to be מחדש בה דבר if built too early and not לשם חג, it would seem that it would only be a d'rabonon requirement, not d'oraysa. The gemara is clear that Beis Hillel rejects the pesukim that Beis Shamai would use to indicate that a Succah needs to be built לשם חג. Therefore, even Beis Hillel would have no Torah source to demand being מחדש בה דבר, so apparently it would only be d'rabonon. The Mishna Berura (636:4) explains that the din of being מחדש בה דבר is not even an absolute din d'rabonon, it is just a mitzvah lichatchila so that one does not come to use something made as their permanent residence as their succah which would be a p'sul d'oraysa.
According to Tosafos we would require even by סוכת גנב"ך which was made for shade but not for Succos, that they be מחדש בה דבר. However, what would be the halacha for סוכת גנב"ך that is made within 30 days of Succos? How does the concept of סתמא לשמה work when it is not even being made by those who are obligated in the mitzvah? Would it even work for Succos that are built by those who are not obligated in the mitzvah and never heard of the mitzvah?
This would depend on how we understand the concept of סתמא לשמה. If the concept is based on the assumption that any Succah built within 30 days of Succos is assumed to be built for succos so even if the person didn't have explicit intent, we are confident that if we were to ask him he would confirm that would have been his intention. According to this approach, a Succah built by a goy even within 30 days one would need to be mechadesh something because for a goy there is no סתמא לשמה. But if we understand that concept similar to how the Steipler at the beginning of Zevachim and Dibros Moshe (Gittin heara 126 - pg. 544) in the context of a korban being סתמא לשמה, that there is an automatic infusion of לשמה since the korban is designated for that purpose, here too we can say that within 30 days of Succos it is automatically infused with lishma. In other words, is the סתמא לשמה a din in the gavra that we assume his subconsciousness would want it to be lishma (which would not apply for a goy) or is it a din in the z'man that anything built during that time assumes a status of built for the chag (which would apply even if a goy builds it). By Rashi citing the concept of שואלין ודורשין קודם לחג ל' יום as the reason we assume that within 30 days of succos it is סתמא לשמה, it seem that the latter approach that the z'man is automatically infused with kedusha is correct. Based on this approach it would not be necessary to be mechadesh something even in a סוכת גנב"ך. However, the Mishna Berura (536:4) citing the achronim takes for granted that one would need to be mechadesh something in a סוכת גנב"ך.