The Law of Conservation of Energy states that:
Energy can't be created nor can be destroyed. It only changes from one form to another.
According to this the total energy in a closed system never changes. I was wondering what this constant energy is when the closed system is the whole Universe.
Is there any estimate of what the total energy of the Universe is? If there is, kindly give a reference for it.
Regarding the conservation of energy in the Universe, the questions linked by Qmechanic (Total energy of the Universe, Is the law of conservation of energy still valid?, Is the total energy of the universe constant?, Conservation of Energy in General Relativity) have answers that already address this is some detail. Regarding the total energy content of the Universe, that's relatively straightforward. The Universe is observed to have flat geometry, or very nearly so, which means it must have near-critical energy density. The critical density is simply $3H^2/8\pi G$, and can be derived from the Friedmann equations. To give a number with dimensions:
$$\rho_{\rm crit} = 1.8788\times 10^{-26}\,h^2\,{\rm kg}\,{\rm m}^{-3}$$
You should replace $h^2$ with your preferred value for the Hubble constant (at the time of interest) in units of $100\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$. At the present day $H_0\sim 70\,{\rm km}\,{\rm s}^{-1}\,{\rm Mpc}^{-1}$, so $h\sim0.7$.
The volume of the Universe is a bit of a slippery concept (e.g. this answer of mine), so I'll just leave my answer here with the density, and you can multiply by whatever volume you're interested in to arrive at a total energy content for that volume. Note that the critical density should be interpreted as a density averaged over very large scales (think of a volume enclosing many clusters of galaxies). Of course the density locally may be very different.
In fact, for our Universe, the total mass-energy and angular momentum are undefined and undefinable. In addition, note that the total mass-energy of a system in general relativity cannot be generally defined. There are, however, a few tools one can employ to measure the total mass-energy of a system in the case of asymptotically flat spacetimes. (Which our universe being FLRW-type is not!) The first is the ADM mass, defined by: \begin{equation} M_{ADM} = \frac{1}{16 \pi} \int_{\partial \Sigma_{\infty}} \sqrt{\gamma} \gamma^{jn} \gamma^{im} \left(\gamma_{mn,j} - \gamma_{jn,m}\right) dS_{i}, \end{equation} which requires the space-time to be asymptotically flat. Another tool is the Komar mass: \begin{equation} M_{K} = \frac{1}{4 \pi} \int_{\Sigma} d^3x \sqrt{\gamma} n_{a} J^{a}_{(t)}, \end{equation} which also requires the space-time to have an asymptotically flat region.
The problem of course is that our Universe, or any spatially homogeneous and non-static one, that is, one that does not contain a global time-like Killing vector, is necessarily not asymptotically flat. So, in general, such definitions of mass and energy are ill-defined for our Universe.
