Noether's theorem, energy and time invariance I have studied high school physics but only have a pop-sci kind of understanding of ideas like Noether's theorem so go easy. 
As far as I know, Noether's theorem simply states that any kind symmetry of a physical system is accompanied by a particular conserved quantity.
The normal example of this is that if we consider a system to be time symmetric (the laws that govern the system are the same at all points in time), then energy conservation emerges. There are, however, supposed to be examples of systems where energy is not conserved, implying that the laws that govern the system isn't symmetric. 
My problem is with this idea that laws are not symmetric over time. What does this really mean? Does this mean that the laws literally change over time? Is it not a very basic assumption for the study of physics that the laws that govern the universe never change? If we found systems where laws changed, how could we ever study the laws? 
I believe that my confusion here is down to a misunderstanding of what it means for a system to be time symmetric but maybe it's something else. Thanks for any answers. 
 A: It's a little more involved than you say because the object that possesses the symmetry is the action for the system. The equations that describe the motion, i.e. the laws of physics for this system are then derived from this action using the Euler-Lagrange equation.
But we can describe a very simple example. Suppose you're in your spaceship and you plan a flyby of some large object e.g. the Sun. You start far from the Sun with some velocity $v$, and the Sun's gravity accelerates you inwards with the usual Newtonian equation for gravity:
$$ F = \frac{GMm}{r^2} $$
You swing in towards the Sun, accelerating as you go, then as you move away from the Sun again its gravity slows you back down and you depart with the same velocity $v$ that you started with. Your kinetic energy is unchanged.
However suppose it turns out that Newton's constant $G$ is actually time dependent and is decreasing with time:
$$ F(t) = \frac{G(t) Mm}{r^2} $$
That means $G$, and therefore the force, is lower for your outward journey than your inward journey. You are decelerated less on your outward journey and you leave with more energy than you started. Energy has not been conserved, and it's the time dependence of $G$ that is responsible.
A: When we consider a physical system, the environment denotes everything else that is not part of the system. 
While the system has dynamical variables, which we are trying to solve, the variables of the environment are assumed given.
It often happens that the system is not isolated, and external forces from the environment act on the system. If the environment is changing, theses forces typically acquire explicit time dependence.
Example: Let the physical system be a child on a swing. The environment may e.g. be a parent pushing the swing in a time-dependent fashion. The energy of the child needs not be conserved.
A: Simplest and most common system not symmetrical due to time shift is simply any mechanical system with dissipation. For example two bricks moving one on another with friction. In such circumstances part of energy is dissipated to heat, which is not taken into consideration in purely idealistic mechanical description.
System as a whole of course conserves energy, but its mechanical part ( described via classical mechanics, involving action, Lagrangian,  Jacobi equations etc) doesn't contain part of description which have to be carried via thermodynamics.
In this case Noether theorem does not work.
There is no need for Quantum General Relativity to understand this example and in fact I would suggest to focus rather on two bricks system to understand this stuff. 
A: There is an amazing example which shows the effect of Noether's theorem:  The non-conservation of energy in General Relativity. It actually requires the understanding of this theory but it is so clear that it is very useful as case of demonstration. 
Noether's theorem says (among others) that if in a physical system a translation in time (each moment in time is equivalent to any other) does not change the system, then the energy in this system is conserved. It works for million of cases, but not for the gravitation in the view of Einstein's theory.
A gravitational field is no longer invariant under translations in time because the gravitational field is considered as curvature of space-time, so space and time are in general curved, in particular the curvature can change in time. So moments in time are then no longer really equivalent.  So then Noether's theorem tells us that the energy is not necessarily conserved. 
That might be a rather shocking result, but it can easily be demonstrated in another way: By a coordinate transformation the gravitational effect can be  locally "transformed away" i.e. locally eliminated, so if in one coordinate system an energy budget (including the gravitational energy) of the system is set up, in the other system, where the  gravitational effect is locally "transformed away", the gravitational energy has gone too (so no conservation of the total energy). This is an ultimate consequence of the equivalence principle.  
A: Yes, violating the uniformity of the laws in time (time translation invariance) indeed means that the laws are NOT the same at different times. 
While it's common in physics to assume that the laws are constant in time, this need not be the case in principle. If the laws change in particular ways, we could still describe the laws and calculate their ramifications and hence do science. 
John Rennie gave a good example of this situation - if Newton's universal law of gravity actually read $F = G(1+t) \frac{m_1 m2}{r^2}$, with $t$ the time since the Big Bang (say), then we will have a very-specific law of physics and we could determine its consequences and test them against experiments and so on. We could do physics, even though the law is not time-invariant. 
Now, things get a bit complicated in General Relativity. In GR what "time" is is a bit subtle so you end up with a theory whose basic laws do not explicitly depend on time (like the $t$ above) but that still allows for the effective-laws to change in time when space is expanding. The result is a bit of a mess as to whether energy is or is not conserved. It kinda depends on what you define as 'energy', and, well, it's a mess.
A: 
My problem is with this idea that laws are not symmetric over time. What does this really mean?

It is supposed to mean that Lagrangian function, which encodes interactions between different degrees of freedom of the system and the environment, does not depend explicitly on time. That is often the case and one can view it as thanks to the fact that basic physical laws of motion and interaction of bodies do not change in time. For example, a Lagrangian of a body orbiting Earth can be written as
$$
L = \frac{1}{2}m(v_x^2 +v_y^2 +v_z^2 ) + \frac{GMm}{\sqrt{x^2+y^2+z^2}}.
$$
The fact that gravity does not change in time implies that $G$ does not change in time either; it is constant, the same number for any time we use the formula. If gravity did change in time, the above Lagrangian would not work. Another one, where some part depends on time could work.
But there may be other reason for time to appear in the Lagrangian.
For example, particle acted upon by an EM wave can be modeled by a Lagrangian of type
$$
L(x,\dot x) = \frac{1}{2}m\dot x^2 + q x E_0\sin(\Omega t).
$$
There is a time-dependent term due to harmonic EM wave, because the wave is given as a process in time. But nothing about physical laws of EM theory changing in time is implied.
