In this work, we establish a relation between entanglement entropy and fractal dimension D of generic many-body wave functions, by generalizing the result of Page Phys. Rev. Lett. 71, 1291 (1993) to the case of sparse random pure states (SRPS). These SRPS living in a Hilbert space of size N are defined as normalized vectors with only ND (0≤D≤1) random nonzero elements. For D=1, these states used by Page represent ergodic states at an infinite temperature. However, for 0<D<1, the SRPS are nonergodic and fractal, as they are confined in a vanishing ratio ND/N of the full Hilbert space. Both analytically and numerically, we show that the mean entanglement entropy S1(A) of a subsystem A, with Hilbert space dimension NA, scales as S1¯(A)∼DlnN for small fractal dimensions D, ND<NA. Remarkably, S1¯(A) saturates at its thermal (Page) value at an infinite temperature, S1¯(A)∼lnNA at larger D. Consequently, we provide an example when the entanglement entropy takes an ergodic value even though the wave function is highly nonergodic. Finally, we generalize our results to Renyi entropies Sq(A) with q>1 and to genuine multifractal states and also show that their fluctuations have ergodic behavior in a narrower vicinity of the ergodic state D=1.