One can determine the "shape" of energy, in any given localized region of space, by measuring the gravitational potential within that localized region of space. It's complicated though for a number of reasons, one of which being that energy is relative: it's dependent upon one's choice of reference frame. But ultimately, for a given frame of reference, measuring the gravitational potential or gravitational acceleration can be used as a way to obtain localized energy density.
As an example of another reason why this can be complicated, let's start with something simple, like electrostatic energy. The electrostatic energy W of a given volume can be calculated using
W=\frac{1}{2} \int \rho V d \tau
where,
W is the energy,
ρ is the charge density at any particular location,
V is the electrostatic potential (not to be confused with volume) at any given location,
and dτ is the differential volume (e.g., dτ could be dxdydz for Cartesian coordinates).
This might lead one to believe that the energy is contained exclusively within the charge, since the integrand is zero everywhere where there is no charge (since ρ is only nonzero when the charge density is nonzero).
But wait! there's another way!
W=\frac{\varepsilon_0}{2} \int_{all \ space} E^2 d \tau
where,
W is the energy,
ε0 is the electrical constant (a.k.a., permittivity of free space),
E is the magnitude of the electric field (not to be confused with energy),
and dτ is the differential volume.
Note that this integral must be done for all space, even in places where the charge density is zero!
So which is it? Is the energy in the charge or is it in the field? Well, both approaches will give you the correct answer for the energy. But only the latter (the energy is contained within the field) is compatible with general relativity.
So as a way to measure the "shape" of electrostatic energy, we can say
\frac{\varepsilon_0}{2} E^2 = \mathrm{electrostatic \ energy \ per \ unit \ volume}
which gives you a "shape" of sorts. (Again, E here is the magnitude of the electric field, not to be confused with energy.)
Okay, but what about other sorts of energy like matter? It's complicated for some of the same reasons. You could integrate over the mass density to get the total mass, then plug that into W=mc^2, which would give you the correct answer. Or you could integrate the square of the magnitude of the gravitational acceleration over all space, and get the answer that way.
The method of integrating the gravitational field over all space is more general however, since it's independent of what type of energy is involved. It works for electrical energy, kinetic energy (details may depend on choice of reference frame), matter energy, thermal energy, all forms of energy. So in that respect, gravitational potential, and the magnitude of gravitational acceleration is a measure of energy density: it has a "shape," so to speak, and it can be measured and quantified.
[Edit: All I'm trying to say here is that gravitational fields are linked to variations in energy density. You can measure the gravitational field, and that tells you something about the energy density.]