Here's an excerpt from my writing today. I'm discussing 13:9 and whether it might refer to participation in the temple cultus: "With various strange teachings do not be carried away, for it is good for the heart to be confident in [God’s] grace, not in foods that do not benefit those who walk in them."
... Two considerations, however, point in a different
direction in this instance. The first is
the way the author describes the teaching in question. Most interpreters render xenos in 13:9
as “strange.” It is of course possible that, if the temple had continued to exist, a later Gentile Christian at some point might have described the normal operations of the temple cultus as "strange." The question is whether anyone prior to 70CE would have, Jew or Gentile believer. On the face of it, it does not seem very likely that any mainstream Christian--especially a Christian Jew--would describe the idea of the sacrifices in the Jerusalem temple as a "strange teaching."
Indeed, it seems unlikely that "strange teachings" relating to the Levitical cultus would have arisen while the temple was still standing. In itself, this warning not to participate in miscellaneous and peculiar practices of a Levitical nature is one piece of an overall puzzle that we are arguing points to a post-70CE date for the sermon. We can imagine that, in the absence of a temple, any number of synagogue and other practices might emerge intending to serve in some way as a substitute for the now defunct sanctuary.  It is neither possible nor necessary for us to know exactly what they were, only to recognize the forces that make their existence more than plausible.
A second argument against the exhortation of 13:9 referring to temple sacrifices is the way that it arises in its literary context. It is plausible enough that the central high priestly argument of Hebrews alludes to the operations of the Jerusalem temple in one way or another, even though it only refers explicitly to the wilderness tabernacle. Although 13:9-14 alludes back to those earlier arguments, it seems to bring up a new issue, even if it is related. These verses are part of a collection of miscellaneous instructions in the final chapter, the letter closing to the sermon. This admonition about strange teachings is neither at the beginning of these final exhortations nor is it the final one.
It thus does not arise in a way that makes it seem like the central point of the whole sermon, which it would presumably be if it were about the Jerusalem temple. Indeed, how odd it would be if, after so much generality in the bulk of the sermon, the author only got to the point in the letter conclusion, after the sermon part was already for all intents and purposes over! It seems much more likely that this later instruction is related to the central argument but only in a somewhat peripheral way.
Accordingly, 13:9-14 does not affect our argument that Hebrews never argues against participation in the mainstream Levitical cultus. Rather, its argument always urges the audience to rely positively on Christ as a means of atonement. The audience can be confident in relation to what Christ has done. While the author argues that the Levitical cultus did not take away sins and was ineffective as a means of atonement, he never then proceeds to say, "So do not participate in it."
 Jukka Thoren, Barnabas Lindars***