The powder infrared spectra of a sulfate-bearing calcitic coral sample are recorded at room temperature in attenuated total reflectance (ATR) geometry and at low temperature in transmission geometry. The comparison of ATR spectra recorded with diamond or Ge crystal confirms that the width and the shape of the prominent absorption bands, related to CO3 groups, are dominantly affected by macroscopic electrostatic interactions. In contrast, the temperature-dependence of the position and width of weak and narrow absorption bands, related to CO3 or SO4 groups, can be interpreted in terms of thermal expansion, anharmonic coupling, and local strain fluctuations. The three SO4 stretching bands display a contrasted thermal behavior, the frequency of one of them increasing with temperature. Based on first-principles calculation, this unusual temperature dependence is traced back to the anisotropy of calcite thermal expansion. These results sustain the previously proposed atomic-scale model of sulfate in calcite and attest to a dominantly structural nature of sulfate in the investigated sample. Accordingly, structurally substituted sulfur in deep-sea calcitic bamboo coral could be used as proxy of seawater sulfate.