  In this paper, we study the finite-temperature quantum fluctuation of a
classical liquid induced by the topology of an effective conical spacetime, as
well as by a quasi-periodic boundary condition. The conical spacetime could be
either a disclination or a cosmic string. In this context, we consider a phonon
field representing quantum excitations of the liquid density, which obeys an
effective Klein- Gordon equation with the sound velocity replaced by the light
velocity. We obtain closed analytic expressions for the thermal Hadamard
function, and consequently, the renormalized mean square density fluctuation of
the liquid along with thermodynamics quantities such as internal energy, free
energy, total energy, and entropy densities. We also discuss the limiting
cases, including low and high-temperature regimes, and the situations in which
there are only either the conical spacetime or quasi-periodicity.
