A novel string-inspired gravitational theory in four spacetime dimensions is proposed as a sum of the modified (R+alpha R-2) gravity motivated by the Starobinsky inflation and the leading Bel-Robinson-tensor-squared correction to the gravitational effective action of superstrings/M-theory compactified down to four dimensions. The possible origin of the theory from higher dimensions is revealed. The proposed Starobinsky-Bel-Robinson action has only two free parameters, which makes it suitable for verifiable physical applications in black hole physics, cosmological inflation and Hawking radiation.