Note: although the question title states 'ancient India', the examples which the OP specifically mentioned ("Pallava period or during Vijayanagar period") fall within the Indian classical (Pallava) and medieval / early modern periods (Vijayanagara).
Pallava Dynasty
This dynasty existed from 275 AD to 897 AD. The capital Kanchipuram was a city of some importance:
Kanchipuram grew in importance when the Pallavas of southern Andhra
Pradesh, wary of constant invasions from the north, moved their
capital south to the city in the 6th century....Under the Pallavas,
Kanchipuram flourished as a centre of Hindu and Buddhist learning....a
Chinese traveller who visited Kanchipuram in 640, recorded that the
city was 6 miles (9.7 km) in circumference
Vijayanagara and Bahmani
The Vijayanagara Empire (1336 - 1646) and, to a lesser extent, the Bahmani Sultanate both developed urban centres. In History of Medieval India, 800-1700, Satish Chandra writes:
The Vijayanagara and Bahmani kingdoms dominated India south of the
Vindhyas, for more than 200 years. They not only built magnificent
capitals and cities, and beautified them with many splendid buildings
and promoted arts and letters, but also provided for law and order and
the development of commerce and handicrafts.
Citing KD Morrison and CM Sinopoli (2006) and Michael C. Howard's 'Transnationalism and Society: An Introduction', among other sources, Wikipedia notes on Vijayanagara:
The city rapidly grew from an ancient pilgrimage center in
13th-century, to being founded as a capital of Vijayanagara Empire in
early 14th century, to being a metropolis stretching by some estimates
to 650 square kilometers by early 16th century.[18] It became the
world's second largest city, after Beijing, by about 1500 CE.
Chandra cites Niccolò de' Conti, a merchant and explorer from Venice who wrote after a visit in 1420 to Vijayanagara:
'The circumference of the city is sixty miles, its walls carried up to
the mountaills, and enclose the valleys at their foot .... In this
city there are estimated to be ninety thousand men fit to bear
arms...'
Ancient South India
Some this is covered in the book cited by the OP. Generally, for ancient south India, evidence of large-scale urbanization is lacking, though there were certainly important trade centres and ports. Examples include Uraiyur, Arikamedu, Alagankulam, Madura and Nagarjunakonda (Vijayapuri in ancient times) which was
a renowned centre of Buddhism and Buddhist learning, also considered
to be an important focus in local trade network. An amphitheatre, the
only one of its kind in the sub-continent, was the best example of
public buildings. A magnificent tank reservoir with galleries on three
sides and a pillared pavilion on the fourth was built in the
northwestern corner of the valley. A massive dock-reservoir was seen
to the north of the citadel. A large number of pillared halls have
been excavated, of which one appears to be a public hall. Numerous
masonry cisterns, with ornamented steps and benches with smooth floors
were unearthed
Other sources:
Jairus Banaji, 'Exploring the Economy of Late Antiquity: Selected Essays' (Cambridge University Press, 2016)
K. Rajan, 'Iron Age - Early Historic Transition in South India: an Appraisal' (2014)
