Is it hard to recover $p$ from $k \phi(p)$? Given $k\phi(p)$, is it hard to recover $p$? Here, $p$ is a large prime, $\phi(\cdot)$ is  Euler's totient function and $k$ is an unknown integer.

Or what's the complexity to recover $p$ from $k \phi(p)$.
If $p$ is prime, then $\phi(p) = p-1$; so the question is "given $k(p-1)$, can someone get a good guess of what $p$ might be?"  It is unlikely that the attacker would be able to limit it to one particular value of $p$ (as there are likely to be multiple values of $p$ that are plausible), however the attacker might be able to construct a short list of possibilities; we will consider it success if that attacker can create such a list with the correct value on it somewhere.

The answer would depend on how big $k$ (and $p-1$) is (and, how big the product is).  If either is small (say, no more than 100 or 200 bits), or alternately smooth (that is, consists of prime factors that are small), then ECM would be able to pull out $k$ (or $p-1$), giving the attacker a good guess.  In addition, if both have large factors, but the product is sufficiently small, then NFS would be able to factor the product.  However, if both have large primes as factors, then it is unlikely to be feasible to recover $p$.

However, the above assumes that the attacker has no further information about either $p$ or $k$; if he (for example) also knows the value $pq$ (for a large prime $q$), then it is easy for him to derive $p$; he selects a random value $a$, and computes $gcd( pq, a^{k\phi(p)}-1 \bmod pq )$; with high probability, that's $p$ (assuming $k$ is unrelated to $q$; if the value of $k$ is (say) a multiple of $q-1$, then there are other ways to recover $p$ and $q$.