Hydrodynamical fluctuations of the electron gas comprise low-frequency and long-range stochastic excitations over a steady state of the system. In this work we develop the theory of hydrodynamic fluctuations in the nonequilibrium electron gas which can be described by its electron temperature. For this case the characteristic spatial-time parameters are the electric charge decay time ${\mathrm{\ensuremath{\tau}}}_{\mathit{M}}$, the electron temperature relaxation time ${\mathrm{\ensuremath{\tau}}}_{\mathit{T}}$, and two corresponding diffusion lengths ${\mathit{L}}_{\mathit{M}}$ and ${\mathit{L}}_{\mathit{T}}$, respectively. The spectral densities of the fluctuations are found and investigated for arbitrary relationships between the fluctuation frequency \ensuremath{\omega} and times ${\mathrm{\ensuremath{\tau}}}_{\mathit{M}}$ and ${\mathrm{\ensuremath{\tau}}}_{\mathit{T}}$, for wave vectors q and lengths ${\mathit{L}}_{\mathit{M}}$, ${\mathit{L}}_{\mathit{T}}$. In the first case, we established the effect of the crossover correlation of the electron density \ensuremath{\delta}n(r,t) and temperature \ensuremath{\delta}T(r,t) fluctuations. The cross correlation depends on \ensuremath{\omega} and changes its sign, which indicates the existence of the frequency range of correlation and anticorrelation of \ensuremath{\delta}n(r,t) and \ensuremath{\delta}T(r,t). In the general case, spectral densities of the fluctuations have non-Lorentz form, which also holds with thermal equilibrium. At the equilibrium the cross-correlation effect leads to only redistribution of the fluctuation intensity over the frequency region. Under nonequilibrium conditions the cross correlation also changes the integral intensities of the fluctuations and is directly related to additional kinetic correlation of the hot electrons caused by electron-electron interaction. The results are applied to the calculation of light scattering by electron plasma fluctuations. The cross-correlation effect gives the essential contribution to the cross section of the light scattering. It is shown that the additional kinetic correlation of hot electrons can be directly measured by means of the light-scattering experiment.