This　paper　investigates　the　global　(in　time)　stability　and　asymptotic　behavior　of　solutions　near　equilibrium　to　the　Rayleigh-Benard　problem　of　incompressible　viscoelastic　fluids　within　a　three-dimensional　horizontal　layer　domain.　We　adapt　our　analysis　to　Lagrangian　coordinates　and　demonstrate　the　global　stability　of　solutions　for　some　classes　of　large　initial　perturbations,　whenever　the　elastic　coefficient　is　sufficiently　large　relative　to　the　large　initial　perturbations.　Additionally,　the　solution　exhibits　an　exponential　time　decay　towards　its　equilibrium.　Furthermore,　we　show　that　the　nonlinear　problem,　which　exhibits　algebraic　convergence　rates　with　respect　to　the　elastic　coefficient,　converges　to　the　corresponding　linearized　problem　as　the　elastic　coefficient　approaches　infinity.　Finally,　we　extend　the　aforementioned　results　to　the　case　without　heat　conductivity.　Our　results　provide　insight　into　the　stabilizing　effect　of　elasticity　in　inhibiting　thermal　instability　in　viscoelastic　fluids　and　the　asymptotic　behavior　of　such　systems,　especially　in　the　context　of　large　perturbations.