An interesting benchmark would be the total energy usage of the CPU while loading a page in each browser. The compiler managing to idle waiting for network packets should theoretically allow some of the CPU cores to enter sleep states. (Not all, since others are still busy rendering the page, and at least one is busy doing whatever non-DMA kernel bits are involved with receiving the network packets.)
It should be the case that if the same amount of work is done, then the energy used will be the same. If it takes less work to compile the web assembly, then less energy (holding all other parameters the same). If you have to idle a CPU, then you probably use more energy (holding all other parameters the same) i.e. because you will spend more time and accomplish the same amount of real work (but waste energy on the idled core, albeit waste very little, accomplishing extra, but non-productive, work). Cannot let some other CPU parameter changes as a result of cores being idled (e.g. frequency gets boosted on non-idle cores as a result of dynamic frequency scaling with idle cores) to run this experiment. Thinking about CPU energy use is interesting :)
The real killer with energy usage in browsers is idle wake ups per second. If you've got a lot of tabs open and they're all running timers, waking up, hitting the network, etc. then they're keeping the CPU from going into low power state and thus wasting a lot of energy.
Even though I'd prefer to use Firefox I tend to stick with Safari due to the battery life advantage which really shows when you open a lot of tabs.
That's for max frequency on a given process node. Power scales with voltage squared. But that doesn't say anything about wasted power. And dynamic scaling screws that up in modern chips.
I believe I could summarize things by saying the only way you can really save energy* doing the same work+ is by using a different semiconductor process (either power/leakage-reduction-focused or smaller).
* For serious values of "energy"
+ Where the same work is not always true for a given task, if one optimizes an algorithm
For chips, power scales with voltage squared. Is also true that P=IV (since both are true, these observations cannot be in contradiction). Apparently, for chips, the current must be proportional to voltage also. Glossing over some details, turning on (off) a transistor is the same as charging (discharging) a capacitor. The energy stored on a capacitor is 1/2 C V^2. If you turn on and off the transistor periodically (say with frequency f) you use 1/2 C V^2 energy f times per second (energy per unit time is power). Normally the capacitance is ignored when discussing how power changes because for a given design the capacitance is a fixed quantity.
I think it is linear for frequency and non-linear for voltage, i.e. P~fCV^2. But in many current CPUS, the feature that adjusts frequency also adjusts voltage. That's why I stipulated that, for my comments to be true, such shenanigans as dynamic frequency (and voltage) scaling must be "turned off." I think the OP was asking, what happens to CPU energy if you load the web page with and without the optimized compilation. The OP was interested in core sleep states, but I think that dynamic frequency scaling is a confounding factor. It would be interesting to see the measurements w/ and w/out that feature perhaps.
To increase the frequency, you also have to increase the voltage so that the transistors charge faster, otherwise they won't be able to switch in the shorter time.
