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In this work we consider the generalized zeta function method to obtaintemperature corrections to the vacuum (Casimir) energy density, at zerotemperature, associated with quantum vacuum fluctuations of a scalar fieldsubjected to a helix boundary condition and whose modes propagate in(3+1)-dimensional Euclidean spacetime. We find closed and analyticalexpressions for both the two-point heat kernel function and free energy densityin the massive and massless scalar field cases. In particular, for the masslessscalar field case, we also calculate the thermodynamics quantities internalenergy density and entropy density, with their corresponding high- andlow-temperature limits. We show that the temperature correction term in thefree energy density must suffer a finite renormalization, by subtracting thescalar thermal blackbody radiation contribution, in order to provide thecorrect classical limit at high temperatures. We check that, at lowtemperature, the entropy density vanishes as the temperature goes to zero, inaccordance with the third law of thermodynamics. We also point out that, at lowtemperatures, the dominant term in the free energy and internal energydensities is the vacuum energy density at zero temperature. Finally, we alsoshow that the pressure obeys an equation of state.

Thermal Casimir effect for the scalar field in flat spacetime under a helix boundary condition