The paper investigates the stability of Einstein theory with a cosmological constant Λ. For Λ > 0, stability is established for small fluctuations around the de Sitter background inside the event horizon, while for Λ < 0, stability is demonstrated for all asymptotically anti-de Sitter metrics. The analysis uses conserved flux-integral expressions derived from the symmetries of the background. For Λ > 0, the Killing energy, associated with a timelike Killing vector, is shown to be positive for fluctuations inside the event horizon but not outside. For Λ < 0, supergravity techniques are used to prove the positivity of the Killing energy for all asymptotically anti-de Sitter metrics, establishing stability. The results show that the stability properties of Einstein theory with a cosmological constant are similar to those with Λ = 0, indicating that the cosmological constant does not lead to fundamental instabilities. The paper also discusses the implications of these results for the stability of de Sitter and anti-de Sitter spaces, highlighting the role of the event horizon and Hawking radiation in the case of de Sitter space. The analysis concludes that the presence of a cosmological constant does not exclude the theory from being stable, as the stability properties are similar to those of Λ = 0.The paper investigates the stability of Einstein theory with a cosmological constant Λ. For Λ > 0, stability is established for small fluctuations around the de Sitter background inside the event horizon, while for Λ < 0, stability is demonstrated for all asymptotically anti-de Sitter metrics. The analysis uses conserved flux-integral expressions derived from the symmetries of the background. For Λ > 0, the Killing energy, associated with a timelike Killing vector, is shown to be positive for fluctuations inside the event horizon but not outside. For Λ < 0, supergravity techniques are used to prove the positivity of the Killing energy for all asymptotically anti-de Sitter metrics, establishing stability. The results show that the stability properties of Einstein theory with a cosmological constant are similar to those with Λ = 0, indicating that the cosmological constant does not lead to fundamental instabilities. The paper also discusses the implications of these results for the stability of de Sitter and anti-de Sitter spaces, highlighting the role of the event horizon and Hawking radiation in the case of de Sitter space. The analysis concludes that the presence of a cosmological constant does not exclude the theory from being stable, as the stability properties are similar to those of Λ = 0.