What does general relativity predict for spacetime without energy? I am kind of new to GR but I have been familiar with the concepts for a long time, I am getting used to the mathematics just now. My question is, what would GR predict if we would have an empty universe? No energy whatsoever, only a spacetime continuum?
What I  mean by energy is any type of energy, including mass, so a 0 curvature universe.
 A: Solutions to $G_{\mu\nu}=0$ are called vacuum solutions in GR, it follows mathematically that this happens if and only if the Ricci tensor vanishes, i.e. the solutions are exactly the Ricci flat Lorentzian manifolds. In most known explicit examples only some region is Ricci flat (e.g. around Schwarzschild or Kerr black holes), but some global vacuum solutions without singularities are known. Their existence contradicts the strong Mach principle, which implies that universe without matter (singularities are interpreted as degenerated matter) must be the flat Minkowski spacetime. One example is the Ozsváth–Schücking vacuum, describing a sinusoidal gravitational wave, others are given by a family of Kasner vacua, describing  quaint expanding universes without matter. In Kasner universes the expansion can never be isotropic, in fact if the volume overall is expanding with time at least one spatial direction will be contracting.
Keep in mind however that "only a spacetime continuum" (zero stress-energy and hence Einstein tensor) does not imply "no energy whatsoever", because gravitational field itself can do work, and therefore carries energy. For instance, Ozsváth–Schücking waves transport energy just as electromagnetic waves would. "No energy whatsoever" means that not only $G_{\mu\nu}=0$ but even $\partial_{\alpha}g_{\mu\nu}=0$, i.e. the spacetime is locally flat Minkowski. Even those can be peculiar though, for instance the locally flat Deutsch–Politzer space contains closed timelike curves ("time machines").
A: OP is looking for the vacuum solutions to the Einstein field equations. Including only the cosmological constant $\Lambda$, the EFE become the Lambdavacuum field equations,
$$
R_{\mu\nu}=\left(\frac12R-\Lambda\right)g_{\mu\nu}\tag{1}
$$
with $R_{\mu\nu}$ the Ricci curvature tensor and $R$ the Ricci scalar. Solutions to this depend on the sign/value of $\Lambda$.
For $\Lambda\neq0$, the spacetime must be treated as curved, as (1) does not admit flat spacetime solutions (cf. Padmanabhan (2003), pdf).
Solutions to (1) result in either de Sitter space (for $\Lambda>0$) or anti-de Sitter space (for $\Lambda<0$). See also this NED article (also by Padmanabhan) and this Physics.SE post.
For $\Lambda=0$, the vacuum field equations can be solved with flat Minkowski space, Schwarzschild space or Kerr space (provided we are looking at the space outside some sphere of non-zero radius). Solutions of this case would be singularity free, Ricci flat but not necessarily Riemann flat. See also this Physics.SE post.
A: The Einstein Field equations are $G_{\mu\nu}=8\pi GT_{\mu\nu}$.
An empty universe would be one where $T_{\mu\nu}=0$ The Einstein field equations would than read
$G_{\mu\nu}=R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}=0$.
The 00 term of this (for an FLRW metric) is
$\left(\frac{\dot{a}}{a}\right)^2=\frac{8\pi G}{3}\rho-\frac{\kappa}{a^2}$. 
You say that you want the density of the universe to be 0, so $\rho=0$. Setting the curvature to be flat, positive, or negative ($\kappa = 0/+1/-1$, respectively), gives you three differential equations to solve for the scale factor as a function of time ($a(t)=...$).
