  The Anderson impurity model with a density of states $\rho(\varepsilon)
\propto |\varepsilon|^r$ containing a power-law pseudogap centered on the Fermi
energy ($\varepsilon = 0$) features for $0<r<1$ a Kondo-destruction quantum
critical point (QCP) separating Kondo-screened and local-moment phases. The
observation of mixed valency in quantum critical $\beta$-YbAlB$_4$ has prompted
study of this model away from particle-hole symmetry. The critical spin
response associated with all Kondo destruction QCPs has been shown to be
accompanied, for $r=0.6$ and noninteger occupation of the impurity site, by a
divergence of the local charge susceptibility on both sides of the QCP. In this
work, we use the numerical renormalization-group method to characterize the
Kondo-destruction charge response using five critical exponents, which are
found to assume nontrivial values only for $0.55\lesssim r < 1$. For $0 < r
\lesssim 0.55$, by contrast, the local charge susceptibility shows no
divergence at the QCP, but rather exhibits nonanalytic corrections to a regular
leading behavior. Both the charge critical exponents and the previously
obtained spin critical exponents satisfy a set of scaling relations derived
from an ansatz for the free energy near the QCP. These critical exponents can
all be expressed in terms of just two underlying exponents: the
correlation-length exponent $\nu(r)$ and the gap exponent $\Delta(r)$. The
ansatz predicts a divergent local charge susceptibility for $\nu<2$, which
coincides closely with the observed range $0.55\lesssim r<1$. Many of these
results are argued to generalize to interacting QCPs that have been found in
other quantum impurity models.
