The paper investigates the stability of Einstein theory with a cosmological constant Λ. For Λ > 0, stability is established for small fluctuations inside the event horizon in de Sitter space, and semi-classical stability is analyzed. For Λ < 0, stability is demonstrated for all asymptotically anti-de Sitter metrics using supergravity techniques. The analysis is based on constructing conserved flux-integral expressions associated with the symmetries of the chosen background. The effects of an event horizon, leading to Hawking radiation, are expressed for general field Hamiltonians. The Killing energy, defined as a flux integral over a two-dimensional spatial surface, is shown to be positive for small oscillations in de Sitter space but not for those beyond the event horizon. For anti-de Sitter space, small oscillations are everywhere positive. The Killing energy is also shown to be positive for all asymptotically vanishing fluctuations in anti-de Sitter space, establishing stability. The results indicate that theories with a non-vanishing cosmological constant cannot be excluded on stability grounds.The paper investigates the stability of Einstein theory with a cosmological constant Λ. For Λ > 0, stability is established for small fluctuations inside the event horizon in de Sitter space, and semi-classical stability is analyzed. For Λ < 0, stability is demonstrated for all asymptotically anti-de Sitter metrics using supergravity techniques. The analysis is based on constructing conserved flux-integral expressions associated with the symmetries of the chosen background. The effects of an event horizon, leading to Hawking radiation, are expressed for general field Hamiltonians. The Killing energy, defined as a flux integral over a two-dimensional spatial surface, is shown to be positive for small oscillations in de Sitter space but not for those beyond the event horizon. For anti-de Sitter space, small oscillations are everywhere positive. The Killing energy is also shown to be positive for all asymptotically vanishing fluctuations in anti-de Sitter space, establishing stability. The results indicate that theories with a non-vanishing cosmological constant cannot be excluded on stability grounds.