Why should the Planck constant be a constant throughout all space? Our value for the Planck constant $h$ can be found on experiments on Earth, but how do we know that the Planck constant doesn't change throughout space, for instance it depends weakly upon the curvature of spacetime? As far as I know we have only done experiments to determine the value of $h$ close to the Earth's surface, so I don't think there's any direct evidence to suggest it should be a constant throughout all space, so I proposed the idea to my friends. 
One called me a nut, and said that the value for Planck constant has to be the same throughout all space otherwise the conservation of energy would be violated. He put forward the following situation: Suppose that in one region of space the Planck constant is found to be $1$ in SI units (call this Space $A$), and you have a photon of frequency $1$ Hertz. This photon then traveled to a different region in space where the value for the Planck constant is now $3$ in SI units(call this Space $B$). Hence the photon has gone from having $1$ Joule of energy to $3$ Joules of energy (by the relation $E = hf$). In his own words, a twitch of a finger here can cause an explosion in a different region in space. To this I replied: So what? As far as I know the conservation of energy argument derives from the fact that space looks the same everywhere we look, and if the Planck constant was really different in different areas of space, then that argument shouldn't hold, and conservation of energy shouldn't hold either. Even if I did want to abuse this system by creating a feedback loop where the energy got larger and larger, the point was if I ever tried to extract energy from Space $B$ back to Space $A$ it would become a small amount of energy again since the value for $h$ decreases. 
Another friend called me a nut, because he believed that in sending the photon from Space $A$ to Space $B$, I would violate the Second Law of Thermodynamics by decreasing the overall entropy. He didn't really expand on it, and (I believe) neither of us have the abilities to calculate the actual entropy change, so this point remains unresolved. 
Finally, another friend called me a nut, because he believed that if the value for $h$ was different in different regions of space, then all our calculations ever done in physics dealing with the stars and their brightness and so on and so forth were all incorrect, and that would be troubling. To this I replied that firstly, the changes in the value for $h$ could possibly be very small for 'normal' regions of space, and secondly that there was nothing inherently WRONG about all these other calculations we've done being wrong, in fact we know they are wrong already because we can't account for either dark matter or dark energy! So in my mind this wasn't a really good argument as to why the Planck constant couldn't be variable over space. 
With all that being said, I'm still fairly certain that the Planck constant is a constant over space, because it is called a constant. So is there an error in my previous arguments, or what is the conclusive reason that the Planck constant cannot be variable over space? 
 A: Take the example of a hydrogen atom.
The energy levels in the hydrogen atom are given by
$$ E_n = -2\pi^2 \frac{m_e^4}{n^2 h^2}.$$
The spacing between two energy levels determines the frequency of an emitted photon when a radiative transition is made between them:
$$ h \nu_{n_2\rightarrow n_1} = \frac{2\pi^2 m_e^4}{h^2}\left(\frac{1}{n_2^{2}} - \frac{1}{n_1^{2}}\right). $$
Thus the frequency of the transition will be proportional to $h^{-3}$. Thus if $h$ changes, then the frequency of spectral lines corresponding to atomic transitions would change considerably.
Now, when we look at distant galaxies we can identify spectral lines corresponding to the atomic transitions in hydrogen. As John Rennie says, these are redshifted and so this could be interpreted as a systematic spatial change in Planck's constant with distance from the Earth. However, this shift would have to be the same in all directions - since redshift appears to be very isotropic on large scales - and would thus place the Earth at the "centre of the universe", which historically has always turned out to be a very bad idea. Alternatively we could assume Planck's constant was the same everywhere is space but was changing with time - this would produce an isotropic signal. (NB: Cosmic redshift cannot be explained in this way since there are other phenomena, such as the time dilation of supernova light curves that do not depend on Planck's constant in the same way, that would be inconsistent).
That is not to say that these issues are not being considered. Most effort has been focused on seeing whether the fine structure constant $\alpha$ varies as we look back in time at distant galaxies. The reason for this is, as Count Iblis correctly (in my view) points out, the quest for a variable $h$ is futile, since any phenomenon we might try to measure is actually a function of the dimensionless $\alpha$, which is proportional to $e^2/hc$. We might claim that if either varied, it would produce a variation in $\alpha$, but this would amount to a choice of units and only the variation in $\alpha$ is fundamental.
So, in the example above, the hydrogen energy levels are actually proportional to $\alpha^2/n^2$.  A change in $\alpha$ can be detected because the relativistic fine structure of spectral lines - i.e the separation in energy between lines in a doublet for instance, also depends on $\alpha^2$, but is different in atomic species with different atomic numbers.
There are continuing claims and counter claims (e.g. see Webb et al. 2011;  Kraiselburd et al. 2013) that $\alpha$ variations of 1 part in a million or so may be present in high redshift quasars (corresponding to a fractional change of $\Delta \alpha/\alpha \sim 10^{-16}$ per year) but that the variation may have an angular dependence (i.e. a dependence on space as well as time).
A: You say:

As far as I know we have only done experiments to determine the value of h close to the Earth's surface

but the value of Planck's constant determines the position of spectral lines, so we have measured it all over the universe.
We do see shifts in spectral lines due to the red shift or gravitational time dilation, but these are explained by relativity rather than changes in Planck's constant.
Energy is not (necessarily) conserved in an expanding universe, but that is not an argument for the value of $h$ changing.
A: As pointed out by Michael Duff here, Planck's constant is a mere conversion constant. So, just as Michael Duff puts it, you can just as well ask if the number of liters to the gallon is constant throughout all space.
