We find the Casimir energy, at finite temperature, for acoustic phonons in a graphene sheet suspended over a rectangular trench, and the corresponding Casimir forces are interpreted as correction terms to the built-in tensions of the graphene. We show that these corrections generally break the tensional isotropy of the membrane, and can increase or decrease the membrane tension. We demonstrate that for a narrow rectangular trench with side-lengths in the order of few nanometers and few micrometers, these temperature corrections are expected to be noticeable (10^(−4) N/m) at room temperature. These corrections would be even more considerable by increasing the temperature, and can be applied for adjusting the built-in tension of the graphene. Consequently we introduce a corrected version for the fundamental resonance frequency of the graphene resonator.

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