Numerical simulations of 2-D Rayleigh–Bénard convection are designed to study the development of convection at the base of the cooling lithosphere. A zero temperature is suddenly imposed at the mantle surface, which has initially a homogeneous temperature. The strongly temperature-dependent viscosity fluid is heated from within, in order to balance the internal temperature drift resulting from global fluid cooling. For a while, the lithosphere cools approximatively as a conductive half-space and the lithospheric isotherms depth grows as the square root of age. As instabilities progressively develop at the base of the lithosphere, lithospheric cooling departs from the half-space model. We propose two different parametrizations of the age of the first dripping instability, using boundary layer marginal stability or quantifying the characteristic timescale of the instability exponential growth as a function of the Rayleigh number and of the viscous temperature scale. Both parametrizations account very well for our numerical estimates of onset times, but with a slightly better adjustment of the viscous temperature scale dependence in the second case. The absolute value of the onset time depends on the amplitude and location of initial temperature perturbations within the box and on the initial temperature structure of the thermal boundary layer (TBL). Furthermore, thermal perturbations of finite amplitude located within the lithosphere (such as the ones induced by transform faults, for example) strongly reduce the age of the first dripping instability. However, the onset time parametrization derived from transient cooling experiments well adjusts the lithospheric age of the first drip instability below a lithosphere cooling perpendicularly to the ridge axis. This study emphasizes the role of the initial topography of the lithospheric isotherms on the development of instabilities.