The effect of a trip is to be knocked prone. Oozes can't be tripped. Given the definition of prone, a gelatinous cube can't be prone as that is its default state.
Trip
Some creatures—such as oozes, creatures without legs, and flying creatures—cannot be tripped.
Prone
The character is lying on the ground.
Generally, template describes the situation if there will be any Hit Dice size change . Wizards online Types and Templates defines this as follows (along with changeing Constitution score)
Before applying any Constitution changes, check the Hit Dice and Hit Points entry in the template. Some templates change the number or size of the creature's Hit Dice or both. Some templates change previously acquired Hit Dice and continue to change any additional Hit Dice the creature gains. Most templates, however, change only the creature's racial Hit Dice (that is, the Hit Dice it has before adding any class levels). Most templates are fairly explicit about what happens to the creature's Hit Dice, so just follow the instructions in the template.
When you know the size and number of the creature's Hit Dice, recalculate the creature's hit points using the modifier from the creature's new Constitution modifier for each Hit Die (whatever its size).
In your example, you re-roll hit dice with new hit dice size.
UPDATE:
From D&D 3.5 Monster Manual Hit Dice entry description:
This line gives the creature’s number and type of Hit Dice (the die rolled to generate hit points), and lists any bonus hit points. A parenthetical note gives the average hit points for a creature of the indicated number of Hit Dice.
D&D generally defines base HD and additional HP (CON bonus and other bonuses if available) for creatures, but also offers a pre-calculated value for HP of that creature. So you can either calculate it or use pre-calculated value.
That means, game mechanics leave the final decision to DM (in most cases), and it is up to him. (Such as; in my games, I generally offer players 3/4 of the die they roll for their HP when they level up, and calculate NPC HDs in a similar way.)
Better you discuss that with your DM and all of your players and find a way that suits everybody.
Best Answer
By Rules as Written: yes, he should have his 6 racial HD o top of his 8 barbarian levels.
The main assumptions behind the design choice described in the Monsters as PCs are:
These are the reasons because CR is considered a better measure for a monster PC effectiveness during adventuring than his mere HD, and why you should usually compensate it by another 50% (up to one half his CR) while the group levels up.
These are the RAW, and you can find out the @AceCalhoon's excellent answer to an analogous question.
But, comparing the resulting character with a 10th level, equally equipped half-orc Barbarian we see that something has gone wrong.
CR is not a good measure of a monster effectiveness as a PC: it is a good measure of how many resources he depletes from a canonical party during a typical encounter. Most monsters aren't designed to stay on the scene for more than one fight: that's the PC's role.
In fact, many monster have really nasty racial features that could tilt the scale a lot. Special forms of movement or defense (like natural armor, being incorporeal or invisible, etc.) should be taken into account when using a weird critter as a PC race.
Dungeons and Dragons 3.0 and 3.5 introduced the concepts of LA (level adjustment) and ECL (effective character level).
Level Adjustment is an amount of "fake" levels that must be added to the race's HD in order to confront it with a base race with levels in PC classes.
Effective Character Level is simply the race's base HD plus its LA.
You can find a list of them on Savage Species (a 3.5 manual on the topic of playable monsters). The list can be found online on the relevant page of the D&D Wiki.
As you can see, a minotaur's ECL is 8th (that is: LA +2). So, the 2nd-level Barbarian in the example is supposed to be in par with the iconic 10th-level half-orc Barbarian. The quick and dirty rules provided with Pathfinder's on this matter are totally out of balance.