  Biochemical clocks are essential for virtually all living systems. A
biochemical clock that is isolated from an external periodic signal and
subjected to fluctuations can oscillate coherently only for a finite number of
oscillations. Furthermore, such an autonomous clock can oscillate only if it
consumes free energy. What is the minimum amount of free energy consumption
required for a certain number of coherent oscillations? We conjecture a
universal bound that answers this question. A system that oscillates coherently
for $\mathcal{N}$ oscillations has a minimal free energy cost per oscillation
of $4\pi^2\mathcal{N} k_B T$. Our bound is valid for general finite Markov
processes, is conjectured based on extensive numerical evidence, is illustrated
with numerical simulations of a known model for a biochemical oscillator, and
applies to existing experimental data.

Statistical Mechanics