This may be an interesting question. Is there any inyun of “tosfos yom tov” by the first day of Rosh HaShanah going into the second? Nafkeh mina would be that from whatever time one normally waits till to start cooking for the 2nd day, another 5 min would have to be added for example and not cook during that time.
The question might be interesting, but there does not appear to be any practical ramification.
The reason for this is that the concept of Tosefes Yom Tov only implies a prohibition of melachah (see Tosafos, Megillah 4b). In this case, there is of course in any case a prohibition on labor, and therefore the whole idea of Tosefes will not apply.
See also Shemiras Shabbos Kehilchasah 28 note 170 concerning the prohibition of hachanah during Tosefes Shabbos.
This is not the same as Shabbos that goes into Yom Tov, because there is an extra prohibition of melachah on Shabbos vis-a-vis Yom Tov. For the two days of Yom Tov, the question will not apparently apply.