We consider a compact Lie group with bi-invariant metric, coming from the Killing form. In this paper, we study Einstein warped product space, \(M = M_1 \times_{f_1} M_2\) for the cases, \((i)\) \(M_1\) is a Lie group \((ii)\) \(M_2\) is a Lie group and \((iii)\) both \(M_1\) and \(M_2\) are Lie groups. Moreover, we obtain the conditions for an Einstein warped product of Lie groups to become a simple product manifold. Then, we characterize the warping function for generalized Robertson-Walker spacetime, \((M = I \times_{f_1} G_2, - dt^2 + f_1^2 g_2)\) whose fiber \(G_2\), being semi-simple compact Lie group of \(\dim G_2>2\), having bi-invariant metric, coming from the Killing form.