How much energy does a lion contain?
Wow, as an astrophysicist who has just logged into biology SE for the first time, I didn't think I'd have a question I could immediately answer.

You are correct about the Sun's output, so what about the lion.

If the lion is in its usual passive state, i.e. lying around as shown in your picture, then you would not go far wrong in treating them as black body radiators (well this will give you an upper limit, though the emissivity of human skin is quite high, so it should be a reasonable approximation.). To estimate a power I need a lion's temperature and its surface area.

According to this site the body temperature of a lion is 311.33 Kelvin.

I found a calculator that used the DuBois formula for surface area (for humans) and put in 440 pounds and 7 feet 10 for the weight and "height" of a (male, adult) lion - this returned a surface area of $3.6\ m^2$ (about twice a, male human, so sounds roughly ok).

Now using the Stefan-Boltzmann formula $P = \sigma A T^4$, I get the power output of a "black body" lion to be about 2 kW.

Thus $10^{12}$ lions have a power output of $2\times 10^{15}\ W$, which is 11 orders of magnitude less than the Sun.

But now take the question at its most basic. Compared to the Sun, the lion is a pretty effective power generation unit. The Sun only generates $2\times 10^{-4}\ W/kg$, whereas a lion-based power source weighs in with a massive $10\ W/kg$! 

EDIT: Note that the calculation just assumes the Lion can produce this kind of power output whatever environment it is in. In practice a Lion absorbs a large fraction of this power from its surroundings and its internal metabolism does not need (and probably cannot) supply 2kW. Thus the 2kW should be reduced to some extent, though I'm not sure a simplistic $T^4 - T_{\rm env}^{4}$ calculation can be correct, unless one of you biologists tells me that a Lion's metabolism shuts down once the ambient (African) temperature approaches 311K (I guess in a human a lot of it goes in evaporating sweat?) Whatever, the order of magnitude of the answer is unchanged.