Q:

Is narrow sense heritability heritable?

I was wondering if narrow sense heritability $(h^2 = \frac{V_A}{V_p})$ was heritable itself. $(h^{2'} = \frac{V_A}{V_{h^2}})$ 

A:

Heritability of heritability
Heritability is a property of a population (see here for more info), not of an individual. One can only calculate heritability for a trait if this trait can take a value for every given individual because you need variance among those individuals (as it is made clear from your equations). In other words, the trait you want to calculate the heritability from has to vary from individual to individual.
As we are looking for the heritability of heritability, then heritability would need to be defined at the individual level in order to calculate its heritability. But heritability is not defined at the individual level.
It makes no sense to talk about the heritability of heritability. Just like it makes no sense to talk about the heritability of the fitness variance or heritability of the species geographic range.
Phylogenetic signal for heritability
You might be interested of whether related lineages tend to share similar heritability (for a specific chosen trait). I do not know of any study on the subject, but it seems likely that related species are more likely to experience similar population size and structure and similar selective pressure on a given trait than more distantly related lineages. Per consequence, I would expect that heritability shows a phylogenetic signal. It would be wrong to say that heritability is heritable though.

