  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 ($\sim 10^{-4} N/m$) at the 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.

Mesoscale and Nanoscale Physics