Let a bundle of photons very concentrated in a very small area so that the space-time is curved as a black hole, and the photons can't escape: this is what I call a massless black hole of photons.
Question: what the lifetime of a massless black hole of photons (with its initial energy as parameter)?
While photons can in principle form a black hole, the black hole will not be massless. The mass of the black hole will be related to the energy of the photons that went into it by Einstein's famous equation $E = mc^2$.
The black hole will be a regular black hole, and classically it has an infinite lifetime. Once you include Hawking radiation the black hole will have a finite lifetime, but that lifetime will be no different to a black hole formed by the equivalent amount of massive particles.
In General Relativity we usually make no distinction between mass and energy. The mass/energy is described by an object called the stress-energy tensor, and this is usually written as an energy density. To write the tensor we convert mass to energy using $E = mc^2$.
The mass of a black hole is a surprisingly elusive concept. As it happens there has recently been a question on just this subject here. The Schwarzschild metric is actually a vacuum solution i.e. it describes a spacetime that has no matter or energy present. You cannot point to any place and say "aha, here is the mass!". What we normally think of as the mass of the black hole is more precisely the ADM mass, which is a property of the spacetime geometry as a whole.
To someone outside it looks the same as a regular, massive black hole.
Classically the lifetime is the lifetime of the universe. It might merge with another black hole. It might last forever. It might meet a singularity in a big crunch if the whole universe contracts to a singularity.
If you are worried that it can't decay by Hawking radiation because an infalling negative energy virtual antielectron has no real electron inside to annihilate with then don't worry. It can interact with photons inside and virtual electrons inside and there can be a whole chain of interactions between a positron coming out of one side of the black hole and an electron coming out the other side with a whole spacelike chain (not temporal chain) of interactions with the photons inside. It will still look like a thermal outpouring of particles.
Any problems with firewalls or information are the same for a regular massive black hole and one with photons inside.
