  Because the commonly adopted viewpoint that the Keldysh parameter $\gamma $
determines the dynamical regime in strong field physics has long been
demonstrated to be misleading, one can ask what happens as relevant physical
parameters, such as laser intensity and frequency, are varied while $\gamma$ is
kept fixed. We present results from our one- and fully three-dimensional
quantum simulations of high-order harmonic generation (HHG) from various bound
states of hydrogen with $n$ up to 40, where the laser intensities and the
frequencies are scaled from those for $n=1$ in order to maintain a fixed
Keldysh parameter $\gamma$$< 1$ for all $n$. We find that as we increase $n$
while keeping $\gamma $ fixed, the position of the cut-off scales in well
defined manner. Moreover, a secondary plateau forms with a new cut-off,
splitting the HHG plateau into two regions. First of these sub-plateaus is
composed of lower harmonics, and has a higher yield than the second one. The
latter extends up to the semiclassical $I_p+3.17U_p$ cut-off. We find that this
structure is universal, and the HHG spectra look the same for all $n\gtrsim 10$
when plotted as a function of the scaled harmonic order. We investigate the
$n$-, $l$- and momentum distributions to elucidate the physical mechanism
leading to this universal structure.
