The energy density spectrum is an observable of gravitational waves. Divergence has appeared in the energy density spectra of the scalar induced gravitational waves for different gauge fixings. To resolve the discrepancy, we investigate the gauge invariance of the scalar induced gravitational waves. It is shown that the gauge invariant induced gravitational waves can be obtained by subtracting the fictitious tensor perturbations via introducing the counter term composed of the first order scalar perturbations. The kernel function uniquely determines the energy density spectrum of the scalar induced gravitational waves. We explicitly calculate the gauge invariant kernel functions in the Newtonian gauge and the uniform density gauge, respectively. The discrepancy between the energy density spectra upon the Newtonian gauge and the uniform density gauge is shown to be eliminated in the gauge invariant framework. In fact, the gauge invariant approach is also available to other kinds of gauge fixings.
Comparing a gauge-invariant formulation and a "conventional complete gauge-fixing approach" for $l=0,1$ mode perturbations on the Schwarzschild background spacetime
