You can not run
sc.pp.normalize_total(adata, target_sum=1e4)
sc.pp.log1p(adata)
twice on the same counts. That would corrupt the data and you would get an error when you run sc.pp.highly_variable_genes(adata) afterwards.
If you want to find variable genes on a subset that has already been processed with these steps, you can skip th above two and directly do sc.pp.highly_variable_genes. However keep in mind that (asuming you have removed non-higly variable genes from the object as a part of the standart pipeline) you will be recalculating the highly_variable_genes within the subset of genes you are left with. If you want to find the highly_variable_genes for that subset of cells among all of the genes, you need to bring back the raw as subset = subset.raw.to_adata(). Note that the standart pipeline saves the raw after normalization and log1p so you won't need to do those again. And in fact even if you want to do those again, you won't be able do so as discussed above.
You can try saving raw count to layers too. However keep in mind that raw keeps genes when you remove them from the adata object, layers don't.
You will need to do adata.uns['log1p']["base"] = None every time after you read data because None's are not properly serialized while saving the data. This is a bug! There is a fix pending. See (this scanpy issue: KeyError: 'base'.)[scverse/scanpy#2239]