All I can think of is that Gollum is an astute student of history; during the trek through the Dead Marshes, he takes the role of loremaster, relating some tales of the Great Siege and the Battle of the Last Alliance:
The Dead Marshes. There was a great battle long ago, yes, so they told him when Sméagol was young, when I was young before the Precious came. It was a great battle. Tall Men with long swords, and terrible Elves, and Orcses shrieking. They fought on the plain for days and months at the Black Gates.
The Two Towers Book IV Chapter 2: "The Passage of the Marshes"
'The old fortress, very old, very horrible now. We used to hear tales from the South, when Sméagol was young, long ago. O yes. we used to tell lots of tales in the evening, sitting by the banks of the Great River, in the willow-lands, when the River was younger too, gollum, gollum.' He began to weep and mutter. The hobbits waited patiently.
'Tales out of the South,' Gollum went on again, `about the tall Men with the shining eyes, and their houses like hills of stone, and the silver crown of their King and his White Tree: wonderful tales. They built very tall towers, and one they raised was silver-white, and in it there was a stone like the Moon, and round it were great white walls. O yes, there were many tales about the Tower of the Moon.'
The Two Towers Book IV Chapter 3: "The Black Gate is Closed"
However, I hesitate to say this was a reflection on Gollum himself; by his own admission, these tales were told to him back when he was only Sméagol, merely remembered by him centuries later.
As far as I can tell, Sméagol's previous interest in "roots and beginnings" did not at all reflect in the Gollum personality, except that he liked to hang out in caves.
However, I can say with confidence that Gollum was still curious, at least about one thing:
"What iss he, my preciouss?" whispered Gollum (who always spoke to himself through never having anyone else to speak to). This is what he had come to find out, for he was not really very hungry at the moment, only curious
He was anxious to appear friendly, at any rate for the moment, and until he found out more about the sword and the hobbit, whether he was quite alone really, whether he was good to eat, and whether Gollum was really hungry.
After a while Gollum began to hiss with pleasure to himself: "Is it nice, my preciousss? Is it juicy? Is it scrumptiously crunchable?" He began to peer at Bilbo out of the darkness.
The Hobbit Chapter 5: "Riddles in the Dark"