A graph is a cograph if and only if it has no induced path on 4 vertices. The twin reduction graph of a graph G is denoted by RG. In this article, we describe the Laplacian eigenvalues and eigenvectors of a cograph G using its twin reduction graph RG, and characterize cographs with even and odd integer eigenvalues, respectively. We provide a construction for the Laplacian cospectral cographs. Further, we provide a complete description of the Laplacian spectrum of H-join of graphs when H is a cograph, and then obtain bounds for the algebraic connectivity of such graphs. We also provide some interesting observations on Hadamard diagonalizable cographs.