Let us be clear:By light you mean the electromagnetic spectrum of classical electrodynamics, of which a very small part of frequencies is the visible light of everyday language.
In this classical frame the energy of the wave (page 13-15) can be described, and will depend on the frequency of light.
BUT the underlying framework of all nature is quantum mechanical, as far as our theoretical understanding goes, and the quantum of light is the photon,defined by the frequency of the classical light it makes up by $energy=hν$
As for bodies getting inevitably colder , that is called the black body radiation of all matter. Yes, it is the surrounding heat bath that keeps the temperatures on the surface of the earth, powered by the sun to start with.
Also in general, if all the thermal energy of an object will eventually be converted to light,
not light, thermal energy is infrared radiation, see the plot above.
but only some light gets converted back to thermal energy,
In a heat bath the process goes both ways, infrared radiation keeps the temperature constant. If the object is in the vacuum of space the energy of radiation will be radiated to the vacuum and the object will acquire the temperature of the vacuum of space.
shouldn't the total amount of light in the universe be increasing whereas the temperature should be decreasing?
The number of photons that make up electromagnetic radiation is not a conserved quantity, because photons interact in various (some complicated) ways and their energy can be distributed to an indeterminable number of lower frequency photons. What is a conserved quantity is the total energy in flat space ( speaking in Special Relativity terms here).
Now the universe is a big word, and our observable universe is a very small part of the hypothesized one in the current cosmological model. . It is based on general relativity and energy conservation is not part of it, for various reasons.
In general the entropy, the disorder of the universe , is what would increase with the continuous black body radiation of masses as we see them.