Recent advances in tuning the long-standing strength-ductility tradeoff have drawn attention to high-entropy alloys (HEAs), and the appearance of hexagonal close-packed (hcp) structures has been emphasized. However, few studies have explored the elastic moduli of hcp HEAs, which is of prime importance for improved understanding of the outstanding mechanical properties. In this work, we focus on a set of equiatomic rare-earth-free HEAs with hcp structures, i.e. ScTiZr, ScTiHf, ScZrHf, TiZrHf, and ScTiZrHf, and their thermo-elastic properties are studied using quantum mechanical first-principles methods. It is found that, for all considered HEAs, the hexagonal axial ratio shows a weak dependence on the temperature effect, and the thermal expansion coefficient remains almost unchanged above room temperature. From the calculated temperature-dependent single-crystal elastic constants, we analyzed the mechanical stability, elastic anisotropy, and derived polycrystalline moduli. Results indicate that the present HEAs exhibit rather high elastic isotropy and large elastic softening resistance. The ab initio predicted Young's modulus, shear modulus, and specific modulus do not obey the rule of mixture, which indicates that there exists a strong intrinsic hardening effect in all of the considered HEAs. The calculated results are in good agreement with the available experimental measurements.