Q:

Is there a hypothesis that attempts to explain patterns of species richness along all three energy-related environmental gradients?

My question is related to one of the oldest question in ecology: "What determines global patterns of species richness?". However, I want to focus on one particular part of this question, which has been bothering me for a long time.
Background information
One of the most widely recognized ecological patters on Earth, which is found at most scales and in most biological taxa, is the latitudinal diversity gradient (LDG) -- there are more species in the tropics than in the temperate regions, and the further away you move from the tropics, the fewer species you encounter. Furthermore, such pattern exists not only along the latitudinal gradient, but species richness also covaries with altitude in terrestrial environments and depth in marine environments, showing the same diversity gradient!
It seems to be fair to suggest that energy should somehow underline all these diversity gradients and create some sort of universal mechanism that would ultimately affect all species richness patterns on Earth. Unfortunately, there is no consensus on questions as grand as this one, but I'm looking for hypothesis that would specifically attempt to explain all three gradients together.
Question
Is there a hypothesis that attempts to explain patterns of species richness along all three energy-related environmental gradients together: latitude, altitude and depth? If there is, what it's weakness? If there's no such hypothesis, do we have reasons to believe that such a broad link across the three gradients can exist?
Please note how I'm trying to emphasize that I don't want you to list all the hypotheses that describe LDG only, but rather the three gradients together.

A:

This is a big question and a very active field of research. I'm not deeply into this litterature, but you should look into the the different scaling relationships (often power laws) that have been described on metabolism vs body size, species-area relationships and species richness vs biomass. Also consider that energy-use by species in a community is considered a zero-sum game both in the neutral model (Hubbell, 2001) and in Red Queen models of evolution. You will probably not find a definite answer to your question though, but there are some interesting intersections of ideas out there. As others have suggested you also need to consider and take into account differential historical extinction rates, and how this will influence current patterns (Mittelbach et al. 2007).
Points of entry could be:

Gaston. 2000. Global patterns in biodiversity (essential reading!)
Chave. 2013. The problem of pattern and scale in ecology: what have we
learned in 20 years?
Venevsky & Veneskaia. 2003. Large-scale energetic and landscape factors of vegetation diversity
Mittelbach et al. 2007. Evolution and the latitudinal diversity gradient: speciation, extinction and biogeography
Marquet et al. 2005. Scaling and power-laws in ecological systems

A couple of caveats/ideas though; first of all, it has been argued that the latitudinal gradient of species richness is likely to be due to many different mechanisms in different taxa (see Gaston, 2000). Here you are looking for a hypothesis to explain not only this single pattern, but all three of them, which makes a single explanation even less likely. Second, many non-exclusive explanations have been put forward for the latitudinal gradient. Even if we don't know which ones of these that are the true mechanisms, some are clearly incompatible with being explanations for all the three patterns you mention, while others might be applicable to all of them. For instance, environmental stability in the tropics is one suggested mechanism for the latitudinal gradient, but this doesn't make much sense for a gradient along ocean depth (deep sea has arguably been a more stable environment both over shorter and longer time-frames). It might be relevant for an altitude gradient though. Going through different hypothesised explanations for the latitudinal gradient in this way (from what I know, the best studied of these gradients) could give you a list of ideas that are most interesting to target (maybe somebody has already done this though). 
I hope this helps.

A:

Nice question!
For the concern of plant diversity over altitudinal gradient it has been shown in several studies in the alps and in Andes that the greatest abundance and diversity is not found in high altitude neither in low altitude but in middle altitude. Some argue that the abundance and diversity depends on temperature and human impact that is what explain this pattern. There is a theoretical model that cut the mountains into three patches along the altitudinal gradient and assume that plants - when they reproduce - either send seeds in the same patch or in a nearby patch. Because, the patch in the middle can receive gene flow from both "sides" while the "extremities" can receive gene flow only from one "side", the diversity is greater in the middle patch.
In aquatic communities, there is very few physical barrier to gene flow and therefore, speciation often occur through sympatry (I think!). Depth has an important role in isolating population (allowing allopatric speciation) because of the sensory drive evolution. Some fishes (to talk only about fishes) have opsins (proteins involved in vision) that are adapted to deep water while other fishes have opsins adapted to shallow water. This divergence, allows to push fishes to chose a preferential habitat and prevent gene flow.
High temperatures lead to high mutation rate and high mutation rate leads to high speciation rate (I think).
Because of past glaciation, it has been found that species in southern hemisphere are younger in average than those in northern hemisphere. Therefore, past ecology has to be taken in account when talking about present diversity. Land mass of different size might also impact species diversity differently.
I don't think a theory that explain species diversity along these three gradients might be easily found. I would tend to think these three gradients are important because they are correlated with parameters that influence species diversity in several ways. Those correlated variables are temperature, humidity, light, time of daylight, land mass, connectivity (rivers that increase gene from mountain to valley or surrounding land mass for example), human impact, …
What did you mean "energy related gradient"? Are you talking about temperature? If so, can your question be reduced to "How does temperature influence species diversity"?
Sorry for the missing references!

This wiki article lists pretty well (better than what I did in my above answer!) different hypothesis for explaining latitudinal gradient in species diversity. Some of these hypotheses can be used to describe effect of depth or altitude.

