The class doesn't satisfy a lot of ZFC, it can't even prove that $|X|<2^{|X|}$.
Note that $HPD(\omega)$.
Externally we know that $(2^{|\omega|})^{HPD}$ is countable
I claim that $HPD$ also see that, this is because $V_{\omega+2}$ has a canonical well-ordering for $(\mathcal P(\omega))^{HPD}$ of ordertype $\omega$ (it is even without parameters), simply by intertwining Ackermann coding for parameters with Godel encoding for formulaes (and noting that every $HPD$ subset of $\omega$ must be seen from $V_{\omega+1}$), so $HPD\models |\omega|=|\mathcal P(\omega)|$
While I'm not quite sure the extract reverse mathematics strength of that proposition, it can't be very high.
Note that it imply at the very least that HPD doesn't have $\Delta_0$ separation/replacement without parameters: like @GabeGoldberg stated, for every $X\subseteq \omega$ we have $X'=X\cup\{\omega\}\in HPD$, but most such $X=\{a\in X'\mid a\ne\omega\}\notin HPD$, and $\omega$ is definable without parameters.