Let
n
≥
3
n \ge 3
,
p
∈
(
1
,
+
∞
)
p \in (1, \, +\infty )
be given. Let
Σ
\Sigma
be an
n
n
-dimensional, closed hypersurface in
R
n
+
1
\mathbb {R}^{n+1}
. It is a well known fact that if
Σ
\Sigma
is an Einstein hypersurface with positive scalar curvature, then it is a round sphere. Here we prove that if a hypersurface is almost Einstein in an
L
p
L^p
-sense, then it is
W
2
,
p
W^{2, \, p}
-close to a sphere and we give a quantitative version of this fact.