Homomorphic Evaluation of Pseudo Random Function I'm looking for a special Pseudo Random Function (PRF) $F_k()$ which follows the below.

For fully homomorphic encryption (FHE), $\mathit{FHE}(F_k(x)) = F_k(\mathit{FHE}(x))$.

Or is there any homomorphically evaluable doubling pseudorandom number generator algorithm? (because Goldreich-Goldwasser-Micali PRF construction is based on the doubling pseudorandom number generator.)  

Thanks.
There is an Eurocrypt'15 paper published by Albrecht et al. which creates a cipher minimizing the number of multiplications needed to produce an encryption. This cipher is called LowMC and it can also be used as a PRF with almost same parameters as in the block-cipher - check out LowMC section in this paper.

Fortunately, there is open-source code which evaluates LowMC homomorphically with HElib. To evaluate the cipher homomorphically is to compute $\mathsf{Enc}_{\mathsf{k}}(\mathsf{LowMC}(x))$ from $\mathsf{Enc}_\mathsf{k}(x)$.

So you can plug in the PRF parameters given by Grassi et al. (for 128 bit PRF security) in the script provided by Albrecht and there you have it: LowMC becomes a PRF and your question is solved - compute $\mathsf{Enc}_\mathsf{k}(\mathsf{PRF}(x))$ from $\mathsf{Enc}_\mathsf{k}(x)$. And I think this is the state of the art for computing PRFs in HE land.