We report the results of high-resolution direct numerical simulations of two-dimensional Rayleigh-Bénard convection for Rayleigh numbers up to Ra=1010 in order to study the influence of temperature boundary conditions on turbulent heat transport. Specifically, we considered the extreme cases of fixed heat flux (where the top and bottom boundaries are poor thermal conductors) and fixed temperature (perfectly conducting boundaries). Both cases display identical heat transport at high Rayleigh numbers fitting a power law Nu≈0.138×Ra0.285 with a scaling exponent indistinguishable from 2/7=0.2857… above Ra=107. The overall flow dynamics for both scenarios, in particular, the time averaged temperature profiles, are also indistinguishable at the highest Rayleigh numbers.