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