We investigate the effect of the accelerated expansion of the Universe due to a cosmological constant, $Lambda$, on the cosmic star formation rate. We utilise hydrodynamical simulations from the EAGLE suite, comparing a $Lambda$CDM Universe to an Einstein-de Sitter model with $Lambda=0$. Despite the differences in the rate of growth of structure, we find that dark energy, at its observed value, has negligible impact on star formation in the Universe. We study these effects beyond the present day by allowing the simulations to run forward into the future ($t>13.8$ Gyr). We show that the impact of $Lambda$ becomes significant only when the Universe has already produced most of its stellar mass, only decreasing the total co-moving density of stars ever formed by ${approx}15%$. We develop a simple analytic model for the cosmic star formation rate that captures the suppression due to a cosmological constant. The main reason for the similarity between the models is that feedback from accreting black holes dramatically reduces the cosmic star formation at late times. Interestingly, simulations without feedback from accreting black holes predict an upturn in the cosmic star formation rate for $t>15$ Gyr due to the rejuvenation of massive ($ > 10^{11} mathrm{M}_{odot}$) galaxies. We briefly discuss the implication of the weak dependence of the cosmic star formation on $Lambda$ in the context of the anthropic principle.
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