Short answer:
We can't use the word that here.
Full answer:
Hawking believes that the earth is unlikely to be the only planet ____ life has developed gradually.
Here we are interested in the last part of the sentence. I'll make it shorter, so it's easy to see what is happening:
- Earth is unlikely to be the only planet ____ life has developed gradually.
The second part of this sentence is a relative clause, shown in brackets below:
- Earth is unlikely to be the only planet ____ [ life has developed gradually].
Relative clauses
Relative clauses are very often used to modify nouns:
- That's the restaurant which [ I like ].
- That's the restaurant where [ I saw Barrack Obama ]
In the examples above, the relative clauses modify the word restaurant. Usually there is a relative word at the beginning of the relative clause. The body of the clause comes after this word. In examples (1, 2), the relative words are which and where. The clauses after the wh- words are in brackets, [ ... ]. In restrictive relative clauses like these, we can use the word that instead of the word which.
If you look at the clauses inside the brackets, you will see that they have gaps in:
- That's the restaurant which [ I like __ ].
- That's the restaurant where [ I met Barack Obama __ ]
Notice that both the sentences above use the word restaurant. It doesn't matter what the noun before the relative clause is. In both the sentences above, we can think of a restaurant as being a place. So how do we decide when to use which and when to use where?
Well, the word which is a pronoun. The word where is a locative preposition (some people think of it as a locative adverb). If we need to know whether to use which or where, we can look at the gap in the relative clause. If we can fill it with a pronoun like it, then we need to use the pronoun which in the relative clause. If we can fill it with the locative preposition there, we need to use the locative where. Let's split our examples into two sentences so it is easier to see:
- That's the restaurant. I like __ .
- That's the restaurant. I met Barack Obama __ .
If we fill in those gaps we will see that we need to use it in the first sentence and there in the second one:
- That's the restaurant. I like it .
- That's the restaurant. I met Barack Obama there .
We can't do this the other way round!
- That's the restaurant. *I like there . (ungrammatical)
- That's the restaurant. *I met Barack Obama it . (ungrammatical)
So we understand the sentences like this:
- That's the restaurant which [ I like
it].
- That's the restaurant where [ I saw Barrack Obama
there ]
The Original Poster's example:
Earth is unlikely to be the only planet ____ [ life has developed gradually __ ].
The gap in the relative clause is at the end here. We can only 'fill it in' with there. We couldn't use it:
- Earth is unlikely to be the only planet ____ [ life has developed gradually
there ].
- *Earth is unlikely to be the only planet ____ [ life has developed gradually
it ]
For this reason we need the relative locative word where to go with the locative there.
Grammar Note 1
In actual fact, the Original Poster's sentence could also have a gap before the word life:
- Earth is unlikely to be the only planet ____ [ __ life has developed gradually].
We could fill this gap with a possessive pronoun:
- Earth is unlikely to be the only planet ____ [ its life has developed gradually].
This sentence would give us the impression that maybe there were other planets where life developed fast. If the gap represents a possessive pronoun, we need the possessive relative pronoun whose. Remember we use relative whose for things as well as people:
- Earth is unlikely to be the only planet whose life has developed gradually.
Grammar Note 2
Notice that the word place is unusual. After the word place we can use relative clauses with gaps representing locative phrases and we don't need to use the word where. Instead we can use no relative word at all, or we can use the relative word that:
- That's the place we ate last time
- Dublin is the place I want to live.
- That's the place we saw that incredible film.
- This is the place they kept the prisoner.
We can't do this with other nouns:
- *That's the restaurant we ate last time. (ungrammatical)
- *Dublin is the city I want to live. (ungrammatical)
- *That's the theatre we saw that incredible film. (ungrammatical)
- *This is the dungeon they kept the prisoner. (ungrammatical)