  We analysed 14 observations with kilohertz quasi-periodic oscillations (kHz
QPOs) of the neutron star X-ray binary XTE J1701$-$462, the first source to
show a clear transition between atoll and Z-like behaviour during a single
outburst. We calculated the average cross-spectrum of both atoll and Z-phase
observations of XTE J1701$-$462 between a reference/hard band (6.1 - 25.7 keV)
and a subject/soft band (2.1 - 5.7 keV) to obtain, using a novel technique, the
average time lags of the lower and upper kHz QPOs. During the atoll phase, we
found that at the frequency of the lower kHz QPO the soft photons lag behind
the hard ones by $18 \pm 8$ $\mu$s, whereas during the Z phase the lags are
$33\pm35$ $\mu$s, consistent with zero. This difference in the lags of both
phases suggests that in XTE J1701$-$462, as observed in other sources, the lags
decrease with increasing luminosity. We found that for both the atoll and Z
phase observations the fractional rms amplitude increases with energy up to
$\sim$10 keV and remains more or less constant at higher energies. Since these
changes in the variability of XTE J1701$-$462 occur within the same outburst,
properties like the mass of the neutron star or the inclination of the system
cannot be responsible for the differences in the timing properties of the kHz
QPOs in the atoll and Z phase. Here we suggest that these differences are
driven by a Comptonizing component or corona, possibly oscillating in a coupled
mode with the innermost regions of the accretion disc.
