Is Einstein’s general relativity applicable to early universe? The Lagrangian of general relativity is linear (first order) in Riemann curvature. The cosmological term, if included, is zero order in Riemann curvature. 
From an effective field theory point of view, all symmetry-permitting action terms should be included. Nothing prevents us from adding terms with 2 or more Riemann curvatures. 
These high order terms are usually suppressed at low energies so that they are not relevant under normal circumstances. That being said, they could play an important role at the early stages of our universe involving high energy effects. 
Are these high order terms rightfully taken into account for the prevalent cosmological models (especially the inflationary model which is supposedly applicable to early universe)? 
And for that matter, one can even entertain terms with multiple torsion tensors to account for the spin current's effect (in addition to the energy-momentum current) on space-time for early universe. 
 A: Classically, if you add any more terms like $R^2$, $R_{\mu \nu} \, R^{\mu \nu}$, $R^3$, etc, to the lagrangian density of general relativity, you'll get field equations that are of higher order than 2 in the metric derivatives.  That would imply problems with causality.
The Einstein-Hilbert action (including the $\Lambda$ term) is the only one that give second order field equations for the metric components.
A: To date there have not been any cases where general relativity failed to meet the data. There are one or two things people are looking at very carefully with still some puzzled expressions. But, so far as solid data, GR is still champion.
So such considerations would be pretty speculative. As far as we can tell, we don't need any other gravity theory. It can be pretty discouraging.
Not to say it couldn't be interesting. It could be very much so. But you have some mighty big shoes to fill with existing data. Those higher order terms need to be really seriously suppressed for things like solar system data.
A book you might get on the subject is Clifford Will's book about GR confronting the data.
