You can't use d'hôpital when talking about one hospital.

Le médecin travaille dans un hôpital. -- The doctor works in a hospital.
Le médecin ne travaille pas dans un hôpital. -- The doctor doesn't work in a hospital.

But if you're talking about more hospitals:

Y a-t-il des hôpitaux dans cette ville? -- Are there hospitals in this town?
Il n'y a pas d'hôpitaux ici. -- There are no hospitals here.

With a prepositional phrase the negation is of the phrase - the object of the preposition does not change its format.  (It is a preposition followed by an article, not the other way around)

« Il travaille DANS un hôpital » or « Il travaille À l'hôpital » - « Il ne travaille pas DANS un hôpital » or « Il ne travaille pas À l'hôpital » (In English "He works IN/AT ....." becomes "He doesn't work IN/AT..."

However if you were commenting on his ownership « Il a un hôpital » becomes « Il n'a pas d'hôpital » 

de followed by h muet becomes d'h......;  de followed by vowel becomes d'vowel . . .

The sentence in question in this case related to « un hôpital » - "a hospital", not 'one hospital', in this instance. 

How do either of these lessons indicate that you cannot use « d'hôpital » in the singular?  Why is it not correct to say « il n'y a pas d'hôpital dans cette ville »? Another example - « Il y a un hôpital à Belfort, mais il n'y a pas d'hôpital à Giromagny ».

An explanation of why an answer that suggests these examples are wrong has been marked as the correct answer would be enlightening.

That noted, the question was mainly on the use of « pas de » when « ne .. pas » is followed immediately by a prepositional phrase, and the answer is that you don't use « pas de », as noted above.

Du/de la/de l'/des all become de/d' in negative sentences (French Partitive Articles)

Un/une become de/d' in negative sentences in French (French Indefinite Articles)