  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.
