Hidden-order phase transition in the heavy-fermion superconductor URu$_2$Si$_2$ exhibits the mean-field-like anomaly in temperature dependence of heat capacity. Motivated by this observation, here we explore the impact of the complex order parameter fluctuations on the thermodynamic properties of the hidden order phase. Specifically, we employ the mean-field theory for the hidden order which describes the hidden order parameter by an average of the hexadecapole operator. We compute the gaussian fluctuation corrections to the mean-field theory equations including both the fluctuations due to hidden order as well as antiferromagnetic order parameters. We find that the gaussian fluctuations lead to the smearing of the second-order transition rendering it to become the first-order one. The strength of the first-order transition is weakly dependent on the strength of underlying antiferromagnetic exchange interactions.
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