Energy-momentum relation in SR and GR There is the famous energy-momentum relation 
$$E^2 - p^2 c^2 = m^2 c^4.$$
I thought it is always valid. "Always" means it is valid in general, so in GR and SR. It is something like an elemantary construction. But now I am not sure and if I understand right it is valid only in SR. Right?
Below I write some more calculations and thoughts for a better understanding of my question.

In SR it is very simple to derive the formula. We start with the metric
$g_{\mu\nu} = \eta_{\mu\nu} = {\rm diag}(1,-1,-1,-1)$
and the product
$g_{\mu\nu} p^\mu p^\nu = m^2 c^2$
where $p^\mu = m u^\mu = m \frac{{\rm d} x^\mu}{{\rm d} \tau}$ is the four-momentum. This gives 
$g_{\mu\nu} p^\mu p^\nu = (p^0)^2 + \eta_{ij} p^i p^j = (E/c)^2 - p^2 = m^2 c^2$
with the definitions $p^0 = E/c$ and $\eta_{ij} p^i p^j = {\mathbf{p}}^2 = p^2$. The last equation is equivialent to the above energy-momentum relation.

Now, when we turn to GR and consider the general metric in spherical coordinates
$g_{\mu\nu} = {\rm diag}(g_{00}, g_{11}, -r^2, -r^2 \sin^2 \vartheta)$
gives
$g_{\mu\nu} p^\mu p^\nu = g_{00} (p^0)^2 + g_{ij} p^i p^j = g_{00} (p^0)^2 - p^2 = m^2 c^4$
Here we have to be carefully now with the definition of $p^0$. In nearly all books about GR we find for the above spherical metric the following relation (or equivalent)
$g_{00} m \frac{{\rm d} x^\mu}{{\rm d} \tau} = E/c = \rm const$
which is interpreted as the conservation of energy. In SR we have $g_{00} = \eta_{00} = 1$ and therefore $p^0 = E/c$. But in GR we have the $g_{00}$ coefficent, e.g.
$g_{00} p^0 = E/c$
This would give the following energy-momentum relation
$\frac{E^2}{g_{00}} - p^2 c^2 = m^2 c^4$
Am I missing or do I mix something, e.g the definition of the time-component $p^0$ through an energy?
Or are the calcutions all right and in GR the energy-momentum relation is indeed different compared to the above famous and well known, which seems to be valid only in SR.
I am heavily confused.

My only idea how to resolve this is to redefine the energy in the general case, e.g.
$\sqrt{g_{00}} p^0 = \mathcal{E}/c$
Note the square root compared to the above definition. This would give an epxression analogue to the famous relation
$\mathcal{E}^2 - p^2 c^2 = m^2 c^4$
But $\mathcal{E}$ is not conserved here any more, $\mathcal{E} \neq \rm const$. Instead we have $\sqrt{g_{00}} \mathcal{E} = E = \rm const$.
Is this a problem?
In SR we have simply $E = \mathcal{E}$ and there is no need to worry. So, SR automatically implies that $p^0$ is conserved. But it is not necessary the case in GR, right? Or is this all just a question of definition/convention?
 A: GR is locally the same as SR -- that's one way of stating the equivalence principle. So stuff like this is exactly the same in GR as in SR. There is actually no GR involved in your question. We can use arbitrary coordinate systems in SR. GR happens when there is curvature, which is not relevant here because we're talking about a single vector, which lives in its own neighborhood of spacetime.
A couple of aspects of your presentation are needlessly complicating things. Do yourself a favor and start working in natural units, with $c=1$, so you don't have to keep writing fatcors of $c$ all over the place. Also, there is no special reason to use spherical coordinates here. GR works with any coordinate system. If you want to reduce this to its essence, just do a 1+1-dimensional metric in a coordinate system where the metric is $\operatorname{diag}(g_{00},g_{11})$.
The basic issue here is that we don't just define energy as the 0th component of the energy-momentum vector. As a trivial example of why we wouldn't do this, consider the coordinate system $(x,t)$, where $x$ and $t$ are local Minkowski coordinates, i.e., we just swap the order. In these coordinates, energy would be the 1th component, not the 0th.
In general, energy is the timelike component of the energy-momentum. For an observer with normalized velocity vector $u$, the energy of an object is $u^\mu p_\mu$.
The identity $E^2-p^2=m^2$ is always valid. It equates the norm of a four-vector to a scalar, and equations of that form are valid in all coordinate systems.
