  We construct an approximate expression for the total cross section for the
production of a heavy quark-antiquark pair in hadronic collisions at
next-to-next-to-next-to-leading order (N$^3$LO) in $\alpha_s$. We use a
technique which exploits the analyticity of the Mellin space cross section, and
the information on its singularity structure coming from large N (soft gluon,
Sudakov) and small N (high energy, BFKL) all order resummations, previously
introduced and used in the case of Higgs production. We validate our method by
comparing to available exact results up to NNLO. We find that N$^3$LO
corrections increase the predicted top pair cross section at the LHC by about
4% over the NNLO.
