Is there a cosmological constant $\Lambda$ in 2D? It is well known that the Einstein tensor is trivially 0 in any two dimensions spacetime:
\begin{equation}\tag{1}
G_{\mu \nu}^{(2)} \equiv 0.
\end{equation}
Thus, it is not possible to formulate the Einstein equation when the spacetime dimension is $D = 2$ and introduce the cosmological constant in the usual way:
\begin{equation}\tag{2}
G_{\mu \nu}^{(D)} + \Lambda_D \, g_{\mu \nu}^{(D)} = -\, 8 \pi G_D \, T_{\mu \nu}^{(D)}.
\end{equation}
This equation imposes $\Lambda_2 = 0$ and $G_2 = 0$ when $D = 2$.
So is there a way to introduce a non-trivial cosmological constant in 2 dimensional spacetimes?  Is $\Lambda_2 \equiv 0$ really an unique necessity?
 A: One thing to note first is that pure gravity (i.e. $\Lambda = 0$) is a topological theory. What this means is that on any manifold, the Einstein action $S_E = \int d^2 x \sqrt{|g|} R$ only depends on quantities on the boundary. In fact, due to the Gauss-Bonnet theorem, if your manifold is compact and has no boundary, then
\begin{equation} \tag{1}
\int d^2 x \sqrt{|g|} R  = n \in \mathbb{Z}
\end{equation}
where $n = 2 - 2 g$ is the Euler characteristic of a genus g surface, and is an integer which is importantly independent of the metric's specific values. This manifests itself in the fact that in the equations motion for the $R$ term vanish identically, i.e. that (exercise)
\begin{equation} \tag{2}
\frac{\delta \sqrt{|g|} R}{\delta g^{\mu\nu}} = 0
\end{equation}
So to compute the equations of motion for the cosmological constant, you get 
\begin{equation}\tag{3}
\end{equation}
\begin{split} 
& \frac{\delta \Lambda  \sqrt{|g|}}{\delta g^{\mu\nu}} \\
   &= \Lambda \frac{\delta \sqrt{|g|}}{\delta g^{\mu\nu}} \\
   &= -\Lambda \frac{1}{2} g_{\mu \nu} \sqrt{-g} = 0 
\end{split}
so the equations of motion are $g_{\mu\nu} = 0$ identically everywhere. 
It is worth mentioning that most places where people talk about gravity in 2D is 2D dilaton gravity, where the gravitational field is coupled to an additional scalar $\phi$ with an action looking like
\begin{equation} \tag{4}
S_E = \int d^2x e^{\phi}(R + \Lambda + (\partial \phi)^2 )
\end{equation}
These types of actions have relevance to string theory, and in more recent developments, they are thought to arise as low energy theories of a quantum mechanical model known as the SYK model.
