How much did it cost to attend events at the ancient Roman colosseum?

SHORT ANSWER

1. Entrance to the Colosseum was most likely free.
2. Spectators were sometimes given free food and gifts.
3. Tickets were probably distributed through patronage and / or were obtained through membership of a club or society.
4. Tickets had seat numbers. Seating was according to social status.

WERE THE GAMES FREE?

The general consensus among historians is that the games were most likely free. In The Colosseum, Keith Hopkins and Mary Beard state:

So far as we know spectators did not pay for their tickets; attendance was one of the perks of citizenship.

Two other sources, though, are more definite: this Repubblica newspaper article, while Tickets to the Colosseum says:

The tickets to the Colosseum were free but everyone had to have a ticket or invitation to gain entry.

The article also says that:

The games were extremely popular and as there were never enough tickets to go round there was, no doubt, a black market operating which sold tickets.

Perhaps one reason for the reluctance of many professional historians to state without reservations that entrance was free is that there were many events at the Colosseum, most of which we have no specific information on. Another reason may be that ancient sources don’t explicitly state ‘entrance to the Colosseum is free’ (perhaps because it was taken for granted). Suetonius, though, comes pretty close (referring to Titus):

At the dedication of his amphitheatre and of the baths which were hastily built near it he gave a most magnificent and costly gladiatorial show.

and also this on Domitian:

He constantly gave grand costly entertainments, both in the amphitheatre and in the Circus,

At the very least, it is evident that the spectacles at the Colosseum were heavily subsidized (otherwise, how could slaves have afforded it?). They were a gift from the sponsor, a means of pleasing and placating the masses (‘bread and circuses’ as TheHonRose has already noted). This sponsor was usually the emperor - restrictions were placed on other 'worthies' sponsoring games, partly out of fear that someone might rival the emperor for popularity.

There is no indication in any of the sources that there were tickets or tokens for different times of the day or for different shows (see here for example).

PROJECTING POWER, & GIFTS TO THE PEOPLE

The Colosseum itself was a gift to the people from a Flavian dynasty eager to legitimize itself. The first of the Flavians, Vespasian, placed the Colosseum on what had previously been Nero’s private estate, a calculated populist move – land that was private and restricted now belonged to the people. At the same time, the Colosseum

provided in the centre of Rome a permanent site for political exhibition, confrontation and control... a venue for the cultivation of'privileged visibility' (to use Greenblatt's phrase, cited by Newlands) and the promotion of a divine distance between the emperor (Titus, Domitian) and the Roman people

Source: A. Boyle & W. Dominik, Flavian Rome

One could argue this was reason enough for entrance to the Colosseum to be free. And this was not all: sometimes spectators were literally showered with tokens for prizes, and / or provided with free food.

On the inaugural games sponsored by Titus, there is this from Cassius Dio:

He would throw down into the theatre from aloft little wooden balls variously inscribed, one designating some article of food, another clothing, another a silver vessel or perhaps a gold one, or again horses, pack-animals, cattle or slaves. Those who seized them were to carry them to the dispensers of the bounty, from whom they would receive the article named.

DISTRIBUTION OF TICKETS

Spectators gained entrance with a token which had their seat number on it. How they acquired these tokens is uncertain. R.B.Abrams, in The Colosseum: A History, summarizes thus:

Some scholars have suggested that senators and influential citizens received blocks of tickets and passed them out as patronage; others theorize that certain trade guilds were given tickets for their members.

Hopkins and Beard also suggest both possibilities for the distribution of tokens or tickets:

…how they were distributed is not clear. Given that everything in ancient Rome, ‘free’ or not, had its price, then we should probably imagine that people paid for membership of clubs and societies to which free tickets were issued. Or men of influence, powerful patrons, distributed tickets to their dependents and clients.

STATUS & SEATING

Social status was the key to seating arrangement. The Classical archaeologist Shelby Brown says:

The Roman arena functioned on one level as a display of the social order, from the slaves and the dispossessed in the ring to the powerful provider of the games at the top of the hierarchy. The illustration of social status in the stands at public shows of all kinds was an important aspect of public entertainment.

Katherine Welch, in The Roman Amphitheatre, says that spectators at the Colosseum were seated as follows:

Senators and Vestal Virgins got front-row seats (as did the emperor and his family). Wealthy equestrians sat above. Then came male Roman citizens of the middle classes and finally women and slaves at the top of the auditorium where the view was poorest….This hierarchical division was achieved not only at the vertical level but also at a horizontal level….the most important people…sat closest to the emperor and his family (probably including the women).

Other sources

R. Laurence, Roman Passions: A History of Pleasure in Rome