Is entropy affected by time? I'm quite new to cryptographic systems, and today we discussed entropy at work. Since entropy is used to ensure the difficulty for an attacker to break in, I was wondering if the more time the high entropy value remains the same, the more the system becomes vulnerable.

Is the concept of entropy related with the time (like a logarithm function or else)?
No.  Entropy does not depend upon time.  Entropy is a purely mathematical, probability-based concept.

The entropy of a distribution is some particular function of the distribution.  The entropy of a random variable is the entropy of its distribution.  Formally speaking, it does not make sense to speak of the entropy of a particular value; but we can talk about the entropy of a random process.  For example: suppose I roll a fair 6-sided die, and get 3.  It doesn't make sense to talk about the entropy of the number 3, but I can ask what is the entropy of this random process (the process of rolling a fair 6-sided die to get a random number from 1 to 6) -- it has $\lg 6$ bits of entropy, in particular.  The entropy of a process is the entropy of the distribution on the possible outputs from that process.

Notice how nothing here changes with time.  If I roll a fair 6-sided die today, the entropy of the result is $\lg 6$ bits.  If I roll it tomorrow, the entropy of that will also be $\lg 6$ bits.  If I roll a fair 6-sided die today, write down the answer, and then a decade from now look at what I wrote down and ask about its entropy, the answer will still be $\lg 6$ bits.  The entropy of this random process remains $\lg 6$ bits no matter how long we wait, because its entropy depends upon the distribution of possible outputs from the random process, and that distribution is whatever it is -- it does not depend upon time.

Now in some cases it might be true that, if you use a random secret key, then the longer you use the same key, the greater the chances that an attacker can recover the key.  That might be true.  But it has nothing to do with entropy.  The entropy doesn't change or degrade with time; the entropy of that key is what it is.  All that changes is, the longer you use the key, the more chances the attacker has to try to break it.  For the same amount of entropy, the more times the attacker can try to break it, the better the attacker's odds will be.  So that's one sense in which the attacker's success probability (or the security of the system) might depend upon how long the system is used for -- even though the entropy of the crypto key does not depend upon that.