  Within the time dependent Ginzburg-Landau theory, the effects of the
superconducting fluctuations on the transport properties above the critical
temperature are characterized by a non-zero imaginary part of the relaxation
rate gamma of the order parameter. Here, we evaluate Im gamma for an
anisotropic dispersion relation typical of the high-Tc cuprate superconductors
(HTS), characterized by a proximity to an electronic topological transition
(ETT). We find that Im gamma abruptly changes sign at the ETT as a function of
doping, in agreement with the universal behavior of the HTS. We also find that
an increase of the in-plane anisotropy, as is given by a non-zero value of the
next-nearest to nearest hopping ratio r=t'/t, increases the value of | Im gamma
| close to the ETT, as well as its singular behavior at low temperature,
therefore enhancing the effect of superconducting fluctuations. Such a result
is in qualitative agreement with the available data for the excess Hall
conductivity for several cuprates and cuprate superlattices.
